Page 1
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Graph polynomials and matroid invariants bycounting graph homomorphisms
Delia Garijo1 Andrew Goodall2
Patrice Ossona de Mendez3 Jarik Nesetril2 Guus Regts4
and Lluıs Vena2
1University of Seville
2Charles University, Prague
3CAMS, CNRS/EHESS, Paris
4University of Amsterdam
27 January 2016 Technion, Haifa
Page 2
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Graph polynomialsGraph homomorphisms
Chromatic polynomial
Definition by evaluations at positive integers
k ∈ N, P(G ; k) = #{proper vertex k-colourings of G}.
P(G ; k) =∑
1≤j≤|V (G)|
(−1)jbj(G )k |V (G)|−j
bj(G ) = #{j-subsets of E (G ) containing no broken cycle}.
(−1)|V (G)|P(G ;−1) = #{acyclic orientations of G}uv ∈ E (G ), P(G ; k) = P(G\uv ; k)− P(G/uv ; k)
Page 3
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Graph polynomialsGraph homomorphisms
Chromatic polynomial
Definition by evaluations at positive integers
k ∈ N, P(G ; k) = #{proper vertex k-colourings of G}.
P(G ; k) =∑
1≤j≤|V (G)|
(−1)jbj(G )k |V (G)|−j
bj(G ) = #{j-subsets of E (G ) containing no broken cycle}.
(−1)|V (G)|P(G ;−1) = #{acyclic orientations of G}uv ∈ E (G ), P(G ; k) = P(G\uv ; k)− P(G/uv ; k)
Page 4
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Graph polynomialsGraph homomorphisms
Chromatic polynomial
Definition by evaluations at positive integers
k ∈ N, P(G ; k) = #{proper vertex k-colourings of G}.
P(G ; k) =∑
1≤j≤|V (G)|
(−1)jbj(G )k |V (G)|−j
bj(G ) = #{j-subsets of E (G ) containing no broken cycle}.
(−1)|V (G)|P(G ;−1) = #{acyclic orientations of G}
uv ∈ E (G ), P(G ; k) = P(G\uv ; k)− P(G/uv ; k)
Page 5
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Graph polynomialsGraph homomorphisms
Chromatic polynomial
Definition by evaluations at positive integers
k ∈ N, P(G ; k) = #{proper vertex k-colourings of G}.
P(G ; k) =∑
1≤j≤|V (G)|
(−1)jbj(G )k |V (G)|−j
bj(G ) = #{j-subsets of E (G ) containing no broken cycle}.
(−1)|V (G)|P(G ;−1) = #{acyclic orientations of G}uv ∈ E (G ), P(G ; k) = P(G\uv ; k)− P(G/uv ; k)
Page 6
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Graph polynomialsGraph homomorphisms
Independence polynomial
Definition by coefficients
I (G ; x) =∑
1≤j≤|V (G)|
bj(G )x j ,
bj(G ) = #{independent subsets of V (G ) of size j}.
v ∈ V (G ), I (G ; x) = I (G − v ; x) + xI (G − N[v ]; x)
(Chudnovsky & Seymour, 2006) K1,3 6⊆i G ⇒ I (G ; x) real roots
bj(G )2 ≥ bj−1(G )bj+1(G ), (implies b1, . . . , b|V (G)| unimodal)
Page 7
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Graph polynomialsGraph homomorphisms
Independence polynomial
Definition by coefficients
I (G ; x) =∑
1≤j≤|V (G)|
bj(G )x j ,
bj(G ) = #{independent subsets of V (G ) of size j}.
v ∈ V (G ), I (G ; x) = I (G − v ; x) + xI (G − N[v ]; x)
(Chudnovsky & Seymour, 2006) K1,3 6⊆i G ⇒ I (G ; x) real roots
bj(G )2 ≥ bj−1(G )bj+1(G ), (implies b1, . . . , b|V (G)| unimodal)
Page 8
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Graph polynomialsGraph homomorphisms
Independence polynomial
Definition by coefficients
I (G ; x) =∑
1≤j≤|V (G)|
bj(G )x j ,
bj(G ) = #{independent subsets of V (G ) of size j}.
v ∈ V (G ), I (G ; x) = I (G − v ; x) + xI (G − N[v ]; x)
(Chudnovsky & Seymour, 2006) K1,3 6⊆i G ⇒ I (G ; x) real roots
bj(G )2 ≥ bj−1(G )bj+1(G ), (implies b1, . . . , b|V (G)| unimodal)
Page 9
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Graph polynomialsGraph homomorphisms
Flow polynomial
Definition (Evaluation at positive integers)
k ∈ N, F (G ; k) = #{nowhere-zero Zk -flows of G}.
F (G ; k) =
F (G/e)− F (G\e) e ordinary
0 e a bridge
(k − 1)F (G\e) e a loop
Page 10
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Graph polynomialsGraph homomorphisms
Tutte polynomial
Definition
For graph G = (V ,E ),
T (G ; x , y) =∑
A⊆E(x − 1)r(E)−r(A)(y − 1)|A|−r(A),
where r(A) is the rank of the spanning subgraph (V ,A) of G .
T (G ; x , y) =
T (G/e; x , y) + T (G\e; x , y) e ordinary
xT (G/e; x , y) e a bridge
yT (G\e; x , y) e a loop,
and T (G ; x , y) = 1 if G has no edges.
Page 11
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Graph polynomialsGraph homomorphisms
Tutte polynomial
Definition
For graph G = (V ,E ),
T (G ; x , y) =∑
A⊆E(x − 1)r(E)−r(A)(y − 1)|A|−r(A),
where r(A) is the rank of the spanning subgraph (V ,A) of G .
T (G ; x , y) =
T (G/e; x , y) + T (G\e; x , y) e ordinary
xT (G/e; x , y) e a bridge
yT (G\e; x , y) e a loop,
and T (G ; x , y) = 1 if G has no edges.
Page 12
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Graph polynomialsGraph homomorphisms
Page 13
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Graph polynomialsGraph homomorphisms
Definition
Graphs G ,H.f : V (G )→ V (H) is a homomorphism from G to H ifuv ∈ E (G ) ⇒ f (u)f (v) ∈ E (H).
Definition
H with adjacency matrix (as,t), weight as,t on st ∈ E (H),
hom(G ,H) =∑
f :V (G)→V (H)
∏
uv∈E(G)
af (u),f (v).
hom(G ,H) = #{homomorphisms from G to H}= #{H-colourings of G}
when H simple (as,t ∈ {0, 1}) or multigraph (as,t ∈ N)
Page 14
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Graph polynomialsGraph homomorphisms
Definition
Graphs G ,H.f : V (G )→ V (H) is a homomorphism from G to H ifuv ∈ E (G ) ⇒ f (u)f (v) ∈ E (H).
Definition
H with adjacency matrix (as,t), weight as,t on st ∈ E (H),
hom(G ,H) =∑
f :V (G)→V (H)
∏
uv∈E(G)
af (u),f (v).
hom(G ,H) = #{homomorphisms from G to H}= #{H-colourings of G}
when H simple (as,t ∈ {0, 1}) or multigraph (as,t ∈ N)
Page 15
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Graph polynomialsGraph homomorphisms
Definition
Graphs G ,H.f : V (G )→ V (H) is a homomorphism from G to H ifuv ∈ E (G ) ⇒ f (u)f (v) ∈ E (H).
Definition
H with adjacency matrix (as,t), weight as,t on st ∈ E (H),
hom(G ,H) =∑
f :V (G)→V (H)
∏
uv∈E(G)
af (u),f (v).
hom(G ,H) = #{homomorphisms from G to H}= #{H-colourings of G}
when H simple (as,t ∈ {0, 1}) or multigraph (as,t ∈ N)
Page 16
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
ExamplesStrongly polynomial sequences of graphs
Example 1
bb
bc
b
bc
bb
bc
bc b b b
b
(Kk)
hom(G ,Kk) = P(G ; k)
chromatic polynomial
Page 17
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
ExamplesStrongly polynomial sequences of graphs
Example 1
bb
bc
b
bc
bb
bc
bc b b b
b
(Kk)
hom(G ,Kk) = P(G ; k)
chromatic polynomial
Page 18
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
ExamplesStrongly polynomial sequences of graphs
Problem 1
Which sequences (Hk) of graphs are such that, for all graphs G ,for each k ∈ N we have
hom(G ,Hk) = p(G ; k)
for polynomial p(G )?
Example
For all graphs G , hom(G ,Kk) = P(G ; k) is the evaluation of thechromatic polynomial of G at k .
Page 19
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
ExamplesStrongly polynomial sequences of graphs
Problem 1
Which sequences (Hk) of graphs are such that, for all graphs G ,for each k ∈ N we have
hom(G ,Hk) = p(G ; k)
for polynomial p(G )?
Example
For all graphs G , hom(G ,Kk) = P(G ; k) is the evaluation of thechromatic polynomial of G at k .
Page 20
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
ExamplesStrongly polynomial sequences of graphs
Example 2: add loops
bb
bc
b
bc
bb
bc
bc b b b
b
(K 1k )
hom(G ,K 1k ) = k |V (G)|
Page 21
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
ExamplesStrongly polynomial sequences of graphs
Example 3: add ` loops
bb
bc
b
bc
bb
bc
bc b b b
b
ℓℓ
ℓ ℓ ℓ
ℓℓ
ℓℓ
ℓ
(K `k)
hom(G ,K `k) =
∑
f :V (G)→[k]
`#{uv∈E(G) | f (u)=f (v)}
= kc(G)(`− 1)r(G)T (G ; `−1+k`−1 , `) (Tutte polynomial)
Page 22
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
ExamplesStrongly polynomial sequences of graphs
Example 3: add ` loops
bb
bc
b
bc
bb
bc
bc b b b
b
ℓℓ
ℓ ℓ ℓ
ℓℓ
ℓℓ
ℓ
(K `k)
hom(G ,K `k) =
∑
f :V (G)→[k]
`#{uv∈E(G) | f (u)=f (v)}
= kc(G)(`− 1)r(G)T (G ; `−1+k`−1 , `) (Tutte polynomial)
Page 23
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
ExamplesStrongly polynomial sequences of graphs
Example 4: add loops weight 1− k
bb
bc
b
bc
bb
bc
bc b b b
b
−1
−1 −2 −2
−2
−3
−3−3
−3
(K 1−kk )
hom(G ,K 1−kk ) =
∑
f :V (G)→[k]
(1− k)#{uv∈E(G) | f (u)=f (v)}
= (−1)|E(G)|k |V (G)|F (G ; k) (flow polynomial)
Page 24
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
ExamplesStrongly polynomial sequences of graphs
Example 4: add loops weight 1− k
bb
bc
b
bc
bb
bc
bc b b b
b
−1
−1 −2 −2
−2
−3
−3−3
−3
(K 1−kk )
hom(G ,K 1−kk ) =
∑
f :V (G)→[k]
(1− k)#{uv∈E(G) | f (u)=f (v)}
= (−1)|E(G)|k |V (G)|F (G ; k) (flow polynomial)
Page 25
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
ExamplesStrongly polynomial sequences of graphs
Example 5
b
bc
b
bc b bcb bb
b
bc b
(K 11 + K1,k)
hom(G ,K 11 + K1,k) = I (G ; k)
independence polynomial
Page 26
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
ExamplesStrongly polynomial sequences of graphs
Example 5
b
bc
b
bc b bcb bb
b
bc b
(K 11 + K1,k)
hom(G ,K 11 + K1,k) = I (G ; k)
independence polynomial
Page 27
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
ExamplesStrongly polynomial sequences of graphs
Example 6
b
bcb b b
b
bc
b
bc
b
bc
bc
bb
bc
b
bc
(K�k2 ) = (Qk) (hypercubes)
Proposition (Garijo, G., Nesetril, 2015)
hom(G ,Qk) = p(G ; k , 2k) for bivariate polynomial p(G )
Page 28
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
ExamplesStrongly polynomial sequences of graphs
Example 6
b
bcb b b
b
bc
b
bc
b
bc
bc
bb
bc
b
bc
(K�k2 ) = (Qk) (hypercubes)
Proposition (Garijo, G., Nesetril, 2015)
hom(G ,Qk) = p(G ; k , 2k) for bivariate polynomial p(G )
Page 29
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
ExamplesStrongly polynomial sequences of graphs
Example 7
b b
bc
b
bc
b b
bc
bc
b b b
b
(Ck)
hom(C3,C3) = 6, hom(C3,Ck) = 0 when k = 2 or k ≥ 4
Page 30
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
ExamplesStrongly polynomial sequences of graphs
Definition
(Hk) is strongly polynomial (in k) if ∀G ∃ polynomial p(G ) suchthat hom(G ,Hk) = p(G ; k) for all k ∈ N.
Example
(Kk), (K 1k ) are strongly polynomial
(K `k) is strongly polynomial (in k , `)
(Qk) not strongly polynomial (but polynomial in k and 2k)
(Ck), (Pk) not strongly polynomial (but eventually polynomialin k)
De la Harpe & Jaeger (1995) construct families of strongly polynomial
sequences, extended by Garijo, G. & Nesetril (2015), and further by G.,
Nesetril & Ossona de Mendez (2016) using quantifier-free interpretation
schemes for finite relational structures (digraphs with added unary
relations).
Page 31
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
ExamplesStrongly polynomial sequences of graphs
Definition
(Hk) is strongly polynomial (in k) if ∀G ∃ polynomial p(G ) suchthat hom(G ,Hk) = p(G ; k) for all k ∈ N.
Example
(Kk), (K 1k ) are strongly polynomial
(K `k) is strongly polynomial (in k , `)
(Qk) not strongly polynomial (but polynomial in k and 2k)
(Ck), (Pk) not strongly polynomial (but eventually polynomialin k)
De la Harpe & Jaeger (1995) construct families of strongly polynomial
sequences, extended by Garijo, G. & Nesetril (2015), and further by G.,
Nesetril & Ossona de Mendez (2016) using quantifier-free interpretation
schemes for finite relational structures (digraphs with added unary
relations).
Page 32
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
ExamplesStrongly polynomial sequences of graphs
Definition
(Hk) is strongly polynomial (in k) if ∀G ∃ polynomial p(G ) suchthat hom(G ,Hk) = p(G ; k) for all k ∈ N.
Example
(Kk), (K 1k ) are strongly polynomial
(K `k) is strongly polynomial (in k , `)
(Qk) not strongly polynomial (but polynomial in k and 2k)
(Ck), (Pk) not strongly polynomial (but eventually polynomialin k)
De la Harpe & Jaeger (1995) construct families of strongly polynomial
sequences, extended by Garijo, G. & Nesetril (2015), and further by G.,
Nesetril & Ossona de Mendez (2016) using quantifier-free interpretation
schemes for finite relational structures (digraphs with added unary
relations).
Page 33
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
ExamplesStrongly polynomial sequences of graphs
Definition
(Hk) is strongly polynomial (in k) if ∀G ∃ polynomial p(G ) suchthat hom(G ,Hk) = p(G ; k) for all k ∈ N.
Example
(Kk), (K 1k ) are strongly polynomial
(K `k) is strongly polynomial (in k , `)
(Qk) not strongly polynomial (but polynomial in k and 2k)
(Ck), (Pk) not strongly polynomial (but eventually polynomialin k)
De la Harpe & Jaeger (1995) construct families of strongly polynomial
sequences, extended by Garijo, G. & Nesetril (2015), and further by G.,
Nesetril & Ossona de Mendez (2016) using quantifier-free interpretation
schemes for finite relational structures (digraphs with added unary
relations).
Page 34
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
ExamplesStrongly polynomial sequences of graphs
Definition
(Hk) is strongly polynomial (in k) if ∀G ∃ polynomial p(G ) suchthat hom(G ,Hk) = p(G ; k) for all k ∈ N.
Example
(Kk), (K 1k ) are strongly polynomial
(K `k) is strongly polynomial (in k , `)
(Qk) not strongly polynomial (but polynomial in k and 2k)
(Ck), (Pk) not strongly polynomial (but eventually polynomialin k)
De la Harpe & Jaeger (1995) construct families of strongly polynomial
sequences, extended by Garijo, G. & Nesetril (2015), and further by G.,
Nesetril & Ossona de Mendez (2016) using quantifier-free interpretation
schemes for finite relational structures (digraphs with added unary
relations).
Page 35
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
ExamplesStrongly polynomial sequences of graphs
Page 36
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Gluing 1-labelled graphs
b
b bG1 G2
G1 G2
1
1
1
G1 ⊔G2
G1 ⊔1 G2
Page 37
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Gluing 2-labelled graphs
b
b b
b b
b
G1 G2
G2G1
1 1
1
2 2
2
G1 ⊔G2
G1 ⊔2 G2
Page 38
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Whitney flip
b
b b
b b
b
G1 G2
G2G1
1 1
1
2 2
2
G1 ⊔2 G2
b
b
G2
1
2
GT2
b
b
G2
G1
1
2
G1 ⊔2 GT2
Whitney flip
Page 39
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Whitney 2-isomorphism theorem
Theorem (Whitney, 1933)
Two graphs G and G ′ have the same cycle matroid if and only ifG ′ can be obtained from G by a sequence of operations of thefollowing three types:
(cut) G1 t1 G2 7−→ G1 t G2
(glue) G1 t G2 7−→ G1 t1 G2
(flip) G1 t2 G2 7−→ G1 t2 GT2
Example
Any two forests with the same number of edges have the samecycle matroid.
Page 40
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Whitney 2-isomorphism theorem
Theorem (Whitney, 1933)
Two graphs G and G ′ have the same cycle matroid if and only ifG ′ can be obtained from G by a sequence of operations of thefollowing three types:
(cut) G1 t1 G2 7−→ G1 t G2
(glue) G1 t G2 7−→ G1 t1 G2
(flip) G1 t2 G2 7−→ G1 t2 GT2
Example
Any two forests with the same number of edges have the samecycle matroid.
Page 41
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Proper colourings and 1-gluing
b
b b
b b
automorphism sending
bb to
1-glue
Page 42
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Proper colourings and 1-gluing
b
b b
b b
automorphism sending
bb to
1-cut
P(G1 t1 G2; k) = P(G1 t G2; k)/k =P(G1; k)P(G2; k)
P(K1; k)
Page 43
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Proper colourings and 1-gluing
b
b b
b b
automorphism sending
bb to
1-glue/ 1-cut
Symk transitive on V (Kk)
P(G1 t1 G2; k) = P(G1 t G2; k)/k =P(G1; k)P(G2; k)
P(K1; k)
Page 44
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Proper colourings and 2-gluing
b
bc
automorphism sending
bb to
b
bc
b
bc
b
bc b
bc
b
bc
b
bc
and b bto
b
bcWhitney flip
2-glue
separate 2-cut
flip over
P(G1 t2 G2) = P(G1 t2 GT2 )
Page 45
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Proper colourings and 2-gluing
b
bc
automorphism sending
bb to
b
bc
b
bc
b
bc b
bc
b
bc
b
bc
and b bto
b
bcWhitney flip
2-glue
separate 2-cut
flip over
Symk generously transitive on V (Kk)
P(G1 t2 G2) = P(G1 t2 GT2 )
Page 46
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Chromatic polynomial as a cycle matroid invariant
Proposition
The graph invariant
P(G ; k)
kc(G)=
hom(G ,Kk)
kc(G)
depends just on the cycle matroid of G .
Problem 2
Which graphs H are such that, for a graph G , we havehom(G ,H)
|V (H)|c(G)= p(G )
where p(G ) depends only on the cycle matroid of G?
Page 47
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Chromatic polynomial as a cycle matroid invariant
Proposition
The graph invariant
P(G ; k)
kc(G)=
hom(G ,Kk)
kc(G)
depends just on the cycle matroid of G .
Problem 2
Which graphs H are such that, for a graph G , we havehom(G ,H)
|V (H)|c(G)= p(G )
where p(G ) depends only on the cycle matroid of G?
Page 48
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Definition
The action of a group Γ on a set S is transitive if for each s, t ∈ Sthere is γ ∈ Γ such that sγ = t.The action of a group Γ on a set S is generously transitive if foreach s, t ∈ S there is γ ∈ Γ such that sγ = t and s = tγ.
Theorem (de la Harpe & Jaeger, 1995)
The graph invariant
G 7→ hom(G ,H)
|V (H)|c(G)
depends just on the cycle matroid of G if H has generouslytransitive automorphism group.
Page 49
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Definition
The action of a group Γ on a set S is transitive if for each s, t ∈ Sthere is γ ∈ Γ such that sγ = t.The action of a group Γ on a set S is generously transitive if foreach s, t ∈ S there is γ ∈ Γ such that sγ = t and s = tγ.
Theorem (de la Harpe & Jaeger, 1995)
The graph invariant
G 7→ hom(G ,H)
|V (H)|c(G)
depends just on the cycle matroid of G if H has generouslytransitive automorphism group.
Page 50
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Definition
The action of a group Γ on a set S is transitive if for each s, t ∈ Sthere is γ ∈ Γ such that sγ = t.The action of a group Γ on a set S is generously transitive if foreach s, t ∈ S there is γ ∈ Γ such that sγ = t and s = tγ.
Theorem (G., Regts & Vena, 2016)
The graph invariant
G 7→ hom(G ,H)
|V (H)|c(G)
depends just on the cycle matroid of G only if H has generouslytransitive automorphism group.
Page 51
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Connection matrices
Definition
Graph invariant G 7→ f (G ) has `th connection matrix
( f (G1 t` G2) )G1,G2
For a graph H, orb`(H) equals number of orbits on `-tuples ofvertices of H under the action of Aut(H).H is twin-free if its adjacency matrix has no two rows equal.
Theorem (Lovasz, 2005)
Let H be a twin-free graph. Then the `th connection matrix ofG 7→ hom(G ,H) has rank orb`(H).
Page 52
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Connection matrices
Definition
Graph invariant G 7→ f (G ) has `th connection matrix
( f (G1 t` G2) )G1,G2
For a graph H, orb`(H) equals number of orbits on `-tuples ofvertices of H under the action of Aut(H).H is twin-free if its adjacency matrix has no two rows equal.
Theorem (Lovasz, 2005)
Let H be a twin-free graph. Then the `th connection matrix ofG 7→ hom(G ,H) has rank orb`(H).
Page 53
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Connection matrices
Definition
Graph invariant G 7→ f (G ) has `th connection matrix
( f (G1 t` G2) )G1,G2
For a graph H, orb`(H) equals number of orbits on `-tuples ofvertices of H under the action of Aut(H).H is twin-free if its adjacency matrix has no two rows equal.
Theorem (Lovasz, 2005)
Let H be a twin-free graph. Then the `th connection matrix ofG 7→ hom(G ,H) has rank orb`(H).
Page 54
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Definition
For labelling φ : [`]→ V (H) and `–labelled G ,
homφ(G ,H) =∑
f :V (G)→V (H)f extends φ
∏
uv∈E(G)
af (u),f (v).
For `-labelled G ,
hom(G ,H) =∑
φ:[`]→V (H)
homφ(G ,H).
For `-labelled G1,G2,
homφ(G1 t` G2,H) = homφ(G1,H)homφ(G2,H).
the `th connection matrix of G 7→ hom(G ,H) is
( homφ(G ,H) )Tφ,G ( homφ(G ,H) )φ,G .
Page 55
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Definition
For labelling φ : [`]→ V (H) and `–labelled G ,
homφ(G ,H) =∑
f :V (G)→V (H)f extends φ
∏
uv∈E(G)
af (u),f (v).
For `-labelled G ,
hom(G ,H) =∑
φ:[`]→V (H)
homφ(G ,H).
For `-labelled G1,G2,
homφ(G1 t` G2,H) = homφ(G1,H)homφ(G2,H).
the `th connection matrix of G 7→ hom(G ,H) is
( homφ(G ,H) )Tφ,G ( homφ(G ,H) )φ,G .
Page 56
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Definition
For labelling φ : [`]→ V (H) and `–labelled G ,
homφ(G ,H) =∑
f :V (G)→V (H)f extends φ
∏
uv∈E(G)
af (u),f (v).
For `-labelled G ,
hom(G ,H) =∑
φ:[`]→V (H)
homφ(G ,H).
For `-labelled G1,G2,
homφ(G1 t` G2,H) = homφ(G1,H)homφ(G2,H).
the `th connection matrix of G 7→ hom(G ,H) is
( homφ(G ,H) )Tφ,G ( homφ(G ,H) )φ,G .
Page 57
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Definition
For labelling φ : [`]→ V (H) and `–labelled G ,
homφ(G ,H) =∑
f :V (G)→V (H)f extends φ
∏
uv∈E(G)
af (u),f (v).
For `-labelled G ,
hom(G ,H) =∑
φ:[`]→V (H)
homφ(G ,H).
For `-labelled G1,G2,
homφ(G1 t` G2,H) = homφ(G1,H)homφ(G2,H).
the `th connection matrix of G 7→ hom(G ,H) is
( homφ(G ,H) )Tφ,G ( homφ(G ,H) )φ,G .
Page 58
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Theorem (Lovasz, 2005)
Let H be a twin-free graph. Then the column space of( homφ(G ,H) )φ,G is precisely the set of vectors invariant underautomorphisms of H.
Proof sketch of our result
Use Lovasz’ theorema and the fact that when hom(G ,H)
|V (H)|c(G) depends
just on the cycle matroid of G the column space of( homφ(G ,H) )φ,G is invariant under (generously) transitive actionof a subgroup of Aut(H). (Taking connection matrices with ` = 1and ` = 2.)
aActually, an extension of it by Guus Regts
Page 59
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Theorem (Lovasz, 2005)
Let H be a twin-free graph. Then the column space of( homφ(G ,H) )φ,G is precisely the set of vectors invariant underautomorphisms of H.
Proof sketch of our result
Use Lovasz’ theorema and the fact that when hom(G ,H)
|V (H)|c(G) depends
just on the cycle matroid of G the column space of( homφ(G ,H) )φ,G is invariant under (generously) transitive actionof a subgroup of Aut(H). (Taking connection matrices with ` = 1and ` = 2.)
aActually, an extension of it by Guus Regts
Page 60
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Page 61
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Proper vertex 3-colouring
b
b b
b b
b
1
1
0
0
2
2
Page 62
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Nowhere-zero Z3-tension
b
b b
b b
b
1
1
0
0
2
2
1
11
2
1 1
1
1
1
Page 63
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Nowhere-zero Z3-tension
b
b
b
b1
11
2
1 1
1
1
1
b
b
+ −
+
+
1 + 1 + 2− 1 = 0 in Z3
Page 64
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Tensions
Graph G with arbitrary orientation of its edges.Traverse edges around a circuit C and let C+ be its forward edgesand C− its backward edges.
Definition
f : E → Zk is a Zk -tension of G if, for each signed circuitC = C+ t C−, ∑
e∈C+
f (e)−∑
e∈C−f (e) = 0.
P(G ; k)
kc(G)= #{nowhere-zero Zk -tensions of G}
Page 65
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Tensions
Graph G with arbitrary orientation of its edges.Traverse edges around a circuit C and let C+ be its forward edgesand C− its backward edges.
Definition
f : E → Zk is a Zk -tension of G if, for each signed circuitC = C+ t C−, ∑
e∈C+
f (e)−∑
e∈C−f (e) = 0.
P(G ; k)
kc(G)= #{nowhere-zero Zk -tensions of G}
Page 66
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Nowhere-zero Z3-flow
b
b b
b b
b
1
1
2
2 2
1
1
1
2
Page 67
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Nowhere-zero Z3-flow
b
b
bb
b1
1
2
2 2 1
1
2
bc1
+
+
−
−
1 + 2− 1− 2 = 0 in Z3
Page 68
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Nowhere-zero Z3-flow
b
bcb1
1
2
2 2
1
2
bc
1
+
+ −−
1
+
bb
1 + 2 + 1− 2− 2 = 0 in Z3
Page 69
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Flows
For a cutset B = E (U,V \U), let B+ be edges of B directed outfrom U to V \U and B− edges of B directed in to U from V \U.
Definition
f : E → Zk is a Zk -flow of G if, for each signed cutsetB = B+ t B−, ∑
e∈B+
f (e)−∑
e∈B−f (e) = 0.
When G planar, circuits in G ∗ are bonds (minimal cutsets) in G .
F (G ; k) = #{nowhere-zero Zk -flows of G}
Page 70
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Flows
For a cutset B = E (U,V \U), let B+ be edges of B directed outfrom U to V \U and B− edges of B directed in to U from V \U.
Definition
f : E → Zk is a Zk -flow of G if, for each signed cutsetB = B+ t B−, ∑
e∈B+
f (e)−∑
e∈B−f (e) = 0.
When G planar, circuits in G ∗ are bonds (minimal cutsets) in G .
F (G ; k) = #{nowhere-zero Zk -flows of G}
Page 71
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Flows
For a cutset B = E (U,V \U), let B+ be edges of B directed outfrom U to V \U and B− edges of B directed in to U from V \U.
Definition
f : E → Zk is a Zk -flow of G if, for each signed cutsetB = B+ t B−, ∑
e∈B+
f (e)−∑
e∈B−f (e) = 0.
When G planar, circuits in G ∗ are bonds (minimal cutsets) in G .
F (G ; k) = #{nowhere-zero Zk -flows of G}
Page 72
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Whitney flip preserves cycle matroid
b
bb
b
b
b
b
bb
b Whitney flipb
b
Page 73
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Whitney flip preserves cycle matroid
b
bb
b
bb Whitney flip
b
b b
b
b
b
+
+
−
−
−
−
+
+
e
f
f
e
edges in flipped half are traversed in reverse order and opposite sign
Page 74
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Tensions
b
bb
bb
b1
11
2
1 2
2
2
2
1 0
22 1
1
b
b b
b b
b
1
1
0
0
2
2
1
11
2
1 1
1
1
1
Whitney flip
Page 75
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Flows
b
b b
b b
b
1
1
2
2 2
1
1
1
2
b
bb
bb
b1
1
2
2 1
2
2
2
2
Whitney flip
Page 76
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Duality
For a planar graph G ,
T (G ∗; x , y) = T (G ; y , x).
For a graph G ,
#{nowhere-zero Zk -tensions} = k−c(G)P(G ; k) = (−1)r(G)T (G ; 1−k , 0),
#{nowhere-zero Zk -flows} = F (G ; k) = (−1)r(G)T (G ; 0, 1− k).
Tutte polynomial extends to any matroid M = (E , r) definedon 2E by size | | and rank function r (or rank/nullity).
Tensions/flows defined for orientable matroids.
Page 77
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Duality
For a planar graph G ,
T (G ∗; x , y) = T (G ; y , x).
For a graph G ,
#{nowhere-zero Zk -tensions} = k−c(G)P(G ; k) = (−1)r(G)T (G ; 1−k , 0),
#{nowhere-zero Zk -flows} = F (G ; k) = (−1)r(G)T (G ; 0, 1− k).
Tutte polynomial extends to any matroid M = (E , r) definedon 2E by size | | and rank function r (or rank/nullity).
Tensions/flows defined for orientable matroids.
Page 78
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Duality
For a planar graph G ,
T (G ∗; x , y) = T (G ; y , x).
For a graph G ,
#{nowhere-zero Zk -tensions} = k−c(G)P(G ; k) = (−1)r(G)T (G ; 1−k , 0),
#{nowhere-zero Zk -flows} = F (G ; k) = (−1)r(G)T (G ; 0, 1− k).
Tutte polynomial extends to any matroid M = (E , r) definedon 2E by size | | and rank function r (or rank/nullity).
Tensions/flows defined for orientable matroids.
Page 79
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Duality
For a planar graph G ,
T (G ∗; x , y) = T (G ; y , x).
For a graph G ,
#{nowhere-zero Zk -tensions} = k−c(G)P(G ; k) = (−1)r(G)T (G ; 1−k , 0),
#{nowhere-zero Zk -flows} = F (G ; k) = (−1)r(G)T (G ; 0, 1− k).
Tutte polynomial extends to any matroid M = (E , r) definedon 2E by size | | and rank function r (or rank/nullity).
Tensions/flows defined for orientable matroids.
Page 80
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
chromatic polynomial
cycle matroid of a graphsigned circuits (from edge orientations)
nowhere-zero tensionscycle matroid of a graph
signed cutsets (from edge orientation)
nowhere-zero flows
orientable matroid
signed circuits/ cocircuits
nowhere-zero tensions/flows
Tutte polynomialmatroid
rank/nullity
Page 81
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
chromatic polynomial
cycle matroid of a graphsigned circuits (from edge orientations)
nowhere-zero tensionscycle matroid of a graph
signed cutsets (from edge orientation)
nowhere-zero flows
orientable matroid
signed circuits/ cocircuits
nowhere-zero tensions/flows
Tutte polynomialmatroid
rank/nullity
P(G;k) = kc(G)(−1)r(G)T(G;1− k,0)
Page 82
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
chromatic polynomial
cycle matroid of a graphsigned circuits (from edge orientations)
nowhere-zero tensionscycle matroid of a graph
signed cutsets (from edge orientation)
nowhere-zero flows
orientable matroid
signed circuits/ cocircuits
nowhere-zero tensions/flows
Tutte polynomialmatroid
rank/nullity
P(G;k) = kc(G)(−1)r(G)T(G;1− k,0)
F(G;k) = (−1)|E|−r(G)T(G;0,1− k)
Page 83
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
chromatic polynomial
cycle matroid of a graphsigned circuits (from edge orientations)
nowhere-zero tensionscycle matroid of a graph
signed cutsets (from edge orientation)
nowhere-zero flows
orientable matroid
signed circuits/ cocircuits
nowhere-zero tensions/flows
Tutte polynomialmatroid
rank/nullity
P(G;k) = kc(G)(−1)r(G)T(G;1− k,0)
F(G;k) = (−1)|E|−r(G)T(G;0,1− k)
T(M;x,y) =∑
A⊆E(x−1)r(E)−r(A)(y−1)|A|−r(A)
Page 84
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
chromatic polynomial
cycle matroid of a graphsigned circuits (from edge orientations)
nowhere-zero tensionscycle matroid of a graph
signed cutsets (from edge orientation)
nowhere-zero flows
orientable matroid
signed circuits/ cocircuits
nowhere-zero tensions/flows
Tutte polynomialmatroid
rank/nullity
P(G;k) = kc(G)(−1)r(G)T(G;1− k,0)
F(G;k) = (−1)|E|−r(G)T(G;0,1− k)
T(M;x,y) =∑
A⊆E(x−1)r(E)−r(A)(y−1)|A|−r(A)
common generalizationto orientable matroids?
Page 85
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Definition
Generalized Johnson graph Jk,r ,D , D ⊆ {0, 1, . . . , r}vertices
([k]r
), edge uv when |u ∩ v | ∈ D
Johnson graphs D = {k − 1} J(k , r)
Kneser graphs D = {0} Kk:r
Petersen graph = K5:2
Figure by Watchduck (a.k.a. Tilman Piesk). Wikimedia Commons
Page 86
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Definition
Generalized Johnson graph Jk,r ,D , D ⊆ {0, 1, . . . , r}vertices
([k]r
), edge uv when |u ∩ v | ∈ D
Johnson graphs D = {k − 1} J(k , r)
Kneser graphs D = {0} Kk:r
Petersen graph = K5:2
Figure by Watchduck (a.k.a. Tilman Piesk). Wikimedia Commons
Page 87
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Definition
Generalized Johnson graph Jk,r ,D , D ⊆ {0, 1, . . . , r}vertices
([k]r
), edge uv when |u ∩ v | ∈ D
Johnson graphs D = {k − 1} J(k , r)
Kneser graphs D = {0} Kk:r
Petersen graph = K5:2
Figure by Watchduck (a.k.a. Tilman Piesk). Wikimedia Commons
Page 88
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Johnson graph J(5, 2)Figure by Watchduck (a.k.a. Tilman Piesk). Wikimedia Commons
Page 89
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Fractional chromatic number of graph G :
χf (G ) = inf{kr
: k , r ∈ N, hom(G ,Kk:r ) > 0},
For k ≥ 2r , χ(Kk:r ) = k − 2r + 2 , while χf (Kk:r ) = kr
Page 90
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Fractional chromatic number of graph G :
χf (G ) = inf{kr
: k , r ∈ N, hom(G ,Kk:r ) > 0},
For k ≥ 2r , χ(Kk:r ) = k − 2r + 2 , while χf (Kk:r ) = kr
Page 91
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Fractional colouring example: C5 to Kk :r
b
bc
b c
bc
b
1
2 2
1 3
b
bc
bc
bc
b
14
25 25
14 36
b
bc
bc
bc
b
14
25 23
34 15
k = 3, r = 1 k = 6, r = 2 k = 5, r = 2
χ(C5) = 3 but by the homomorphism from C5 to Kneser graphK5:2 (Petersen graph) χf (C5) ≤ 5
2 (in fact with equality)
Page 92
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Proposition
For a graph G and k, r ≥ 1,hom(G ,Kk:r ) = (r !)−|V (G)|P(G [Kr ]; k).
Proposition (de la Harpe & Jaeger, 1995; Garijo, G., Nesetril, 2015)
For every r ,D, sequence (Jk,r ,D) is strongly polynomial (in k).
Proposition (de la Harpe & Jaeger, 1995)
The graph parameter(kr
)−c(G)hom(G , Jk,r ,D) depends only on the
cycle matroid of G .
Problem
Interpret(kr
)−c(G)hom(G , Jk,r ,D) in terms of the cycle matroid of
G alone. E.g what is its evaluation at k = −1 (acyclic orientationsfor the chromatic polynomial = 1,D = {0}).
Page 93
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Proposition
For a graph G and k, r ≥ 1,hom(G ,Kk:r ) = (r !)−|V (G)|P(G [Kr ]; k).
Proposition (de la Harpe & Jaeger, 1995; Garijo, G., Nesetril, 2015)
For every r ,D, sequence (Jk,r ,D) is strongly polynomial (in k).
Proposition (de la Harpe & Jaeger, 1995)
The graph parameter(kr
)−c(G)hom(G , Jk,r ,D) depends only on the
cycle matroid of G .
Problem
Interpret(kr
)−c(G)hom(G , Jk,r ,D) in terms of the cycle matroid of
G alone. E.g what is its evaluation at k = −1 (acyclic orientationsfor the chromatic polynomial = 1,D = {0}).
Page 94
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Proposition
For a graph G and k, r ≥ 1,hom(G ,Kk:r ) = (r !)−|V (G)|P(G [Kr ]; k).
Proposition (de la Harpe & Jaeger, 1995; Garijo, G., Nesetril, 2015)
For every r ,D, sequence (Jk,r ,D) is strongly polynomial (in k).
Proposition (de la Harpe & Jaeger, 1995)
The graph parameter(kr
)−c(G)hom(G , Jk,r ,D) depends only on the
cycle matroid of G .
Problem
Interpret(kr
)−c(G)hom(G , Jk,r ,D) in terms of the cycle matroid of
G alone. E.g what is its evaluation at k = −1 (acyclic orientationsfor the chromatic polynomial = 1,D = {0}).
Page 95
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Proposition
For a graph G and k, r ≥ 1,hom(G ,Kk:r ) = (r !)−|V (G)|P(G [Kr ]; k).
Proposition (de la Harpe & Jaeger, 1995; Garijo, G., Nesetril, 2015)
For every r ,D, sequence (Jk,r ,D) is strongly polynomial (in k).
Proposition (de la Harpe & Jaeger, 1995)
The graph parameter(kr
)−c(G)hom(G , Jk,r ,D) depends only on the
cycle matroid of G .
Problem
Interpret(kr
)−c(G)hom(G , Jk,r ,D) in terms of the cycle matroid of
G alone.
E.g what is its evaluation at k = −1 (acyclic orientationsfor the chromatic polynomial = 1,D = {0}).
Page 96
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Proposition
For a graph G and k, r ≥ 1,hom(G ,Kk:r ) = (r !)−|V (G)|P(G [Kr ]; k).
Proposition (de la Harpe & Jaeger, 1995; Garijo, G., Nesetril, 2015)
For every r ,D, sequence (Jk,r ,D) is strongly polynomial (in k).
Proposition (de la Harpe & Jaeger, 1995)
The graph parameter(kr
)−c(G)hom(G , Jk,r ,D) depends only on the
cycle matroid of G .
Problem
Interpret(kr
)−c(G)hom(G , Jk,r ,D) in terms of the cycle matroid of
G alone. E.g what is its evaluation at k = −1 (acyclic orientationsfor the chromatic polynomial = 1,D = {0}).
Page 97
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
chromatic polynomial
cycle matroid of a graphsigned circuits (from edge orientations)
nowhere-zero tensionscycle matroid of a graph
signed cutsets (from edge orientation)
nowhere-zero flows
orientable matroid
signed circuits/ cocircuits
nowhere-zero tensions/flows
Tutte polynomialmatroid
rank/nullity
proper vertex colourings
P (G; k)
k−c(G)P (G; k) F (G; k)
?
T (M ;x, y)
Page 98
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
chromatic polynomial
cycle matroid of a graphsigned circuits (from edge orientations)
nowhere-zero tensionscycle matroid of a graph
signed cutsets (from edge orientation)
nowhere-zero flows
orientable matroid
signed circuits/ cocircuits
nowhere-zero tensions/flows
Tutte polynomialmatroid
rank/nullity
proper vertex colourings
P (G; k)
k−c(G)P (G; k) F (G; k)
?
T (M ;x, y)
Kneser/Johnson colouring
s
hom(G, Jk,r,D)
a Kneser/Johnson Tutte polynomial?
dual to Kneser/Johnson colouring
s?
Page 99
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Current thoughts...
cycle matroid of a graph
circuit rotations (graph traversal)
non-Abelian tensions?cycle matroid of a graph
signed cutsets (from edge orientation)
non-Abelian flows?
“rotatable” orientable matroid?
signed/rotated circuits/ cocircuits?
nowhere-zero tensions/flows
?
delta matroid?
Kneser/Johnson colourings
(kr
)−c(G)hom(G, Jk,r,D)
?
?
hom(G, Jk,r,D)
signed circuits (from edge orientations)
vertex rotations (orientable embedding)
Page 100
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Gluing product of graphsMain resultTensions and flowsFrom proper to fractional colourings and beyond
Page 101
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
I When is hom(G ,Cayley(Ak ,Bk)) a fixed polynomial(dependent on G ) in |Ak |, |Bk |, where Bk = −Bk ⊆ Ak?
(hypercubes) hom(G ,Cayley(Zk2 ,S1)) polynomial in 2k and k
(S1 = {weight 1 vectors}). [Garijo, G., Nesetril 2015]For D ⊂ N, hom(G ,Cayley(Zk ,±D)) is polynomial in k forsufficiently large k iff D is finite or cofinite. [de la Harpe &Jaeger, 1995](circular colourings)hom(G ,Cayley(Zks , {kr , kr+1, . . . , k(s−r)})) polynomial ink. [G., Nesetril, Ossona de Mendez 2015]
I Which graph polynomials defined by strongly polynomialsequences of graphs satisfy a reduction formula(size-decreasing recurrence) like the chromatic polynomial andindependence polynomial?
Page 102
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
I When is hom(G ,Cayley(Ak ,Bk)) a fixed polynomial(dependent on G ) in |Ak |, |Bk |, where Bk = −Bk ⊆ Ak?
(hypercubes) hom(G ,Cayley(Zk2 ,S1)) polynomial in 2k and k
(S1 = {weight 1 vectors}). [Garijo, G., Nesetril 2015]
For D ⊂ N, hom(G ,Cayley(Zk ,±D)) is polynomial in k forsufficiently large k iff D is finite or cofinite. [de la Harpe &Jaeger, 1995](circular colourings)hom(G ,Cayley(Zks , {kr , kr+1, . . . , k(s−r)})) polynomial ink. [G., Nesetril, Ossona de Mendez 2015]
I Which graph polynomials defined by strongly polynomialsequences of graphs satisfy a reduction formula(size-decreasing recurrence) like the chromatic polynomial andindependence polynomial?
Page 103
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
I When is hom(G ,Cayley(Ak ,Bk)) a fixed polynomial(dependent on G ) in |Ak |, |Bk |, where Bk = −Bk ⊆ Ak?
(hypercubes) hom(G ,Cayley(Zk2 ,S1)) polynomial in 2k and k
(S1 = {weight 1 vectors}). [Garijo, G., Nesetril 2015]For D ⊂ N, hom(G ,Cayley(Zk ,±D)) is polynomial in k forsufficiently large k iff D is finite or cofinite. [de la Harpe &Jaeger, 1995]
(circular colourings)hom(G ,Cayley(Zks , {kr , kr+1, . . . , k(s−r)})) polynomial ink. [G., Nesetril, Ossona de Mendez 2015]
I Which graph polynomials defined by strongly polynomialsequences of graphs satisfy a reduction formula(size-decreasing recurrence) like the chromatic polynomial andindependence polynomial?
Page 104
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
I When is hom(G ,Cayley(Ak ,Bk)) a fixed polynomial(dependent on G ) in |Ak |, |Bk |, where Bk = −Bk ⊆ Ak?
(hypercubes) hom(G ,Cayley(Zk2 ,S1)) polynomial in 2k and k
(S1 = {weight 1 vectors}). [Garijo, G., Nesetril 2015]For D ⊂ N, hom(G ,Cayley(Zk ,±D)) is polynomial in k forsufficiently large k iff D is finite or cofinite. [de la Harpe &Jaeger, 1995](circular colourings)hom(G ,Cayley(Zks , {kr , kr+1, . . . , k(s−r)})) polynomial ink . [G., Nesetril, Ossona de Mendez 2015]
I Which graph polynomials defined by strongly polynomialsequences of graphs satisfy a reduction formula(size-decreasing recurrence) like the chromatic polynomial andindependence polynomial?
Page 105
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
I When is hom(G ,Cayley(Ak ,Bk)) a fixed polynomial(dependent on G ) in |Ak |, |Bk |, where Bk = −Bk ⊆ Ak?
(hypercubes) hom(G ,Cayley(Zk2 ,S1)) polynomial in 2k and k
(S1 = {weight 1 vectors}). [Garijo, G., Nesetril 2015]For D ⊂ N, hom(G ,Cayley(Zk ,±D)) is polynomial in k forsufficiently large k iff D is finite or cofinite. [de la Harpe &Jaeger, 1995](circular colourings)hom(G ,Cayley(Zks , {kr , kr+1, . . . , k(s−r)})) polynomial ink . [G., Nesetril, Ossona de Mendez 2015]
I Which graph polynomials defined by strongly polynomialsequences of graphs satisfy a reduction formula(size-decreasing recurrence) like the chromatic polynomial andindependence polynomial?
Page 106
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
I When is hom(G ,Cayley(Ak ,Bk)) a fixed polynomial(dependent on G ) in |Ak |, |Bk |, where Bk = −Bk ⊆ Ak?
(hypercubes) hom(G ,Cayley(Zk2 ,S1)) polynomial in 2k and k
(S1 = {weight 1 vectors}). [Garijo, G., Nesetril 2015]For D ⊂ N, hom(G ,Cayley(Zk ,±D)) is polynomial in k forsufficiently large k iff D is finite or cofinite. [de la Harpe &Jaeger, 1995](circular colourings)hom(G ,Cayley(Zks , {kr , kr+1, . . . , k(s−r)})) polynomial ink . [G., Nesetril, Ossona de Mendez 2015]
I Which graph polynomials defined by strongly polynomialsequences of graphs satisfy a reduction formula(size-decreasing recurrence) like the chromatic polynomial andindependence polynomial?
Page 107
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Page 108
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Beyond polynomials? Rational generating functions
I For strongly polynomial sequence (Hk),
∑
k
hom(G ,Hk)tk =PG (t)
(1− t)|V (G)|+1
with polynomial PG (t) of degree at most |V (G )|.
I For eventually polynomial sequence (Hk) such as (Ck),
∑
k
hom(G ,Hk)tk =PG (t)
(1− t)|V (G)|+1
with polynomial PG (t).
Page 109
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Beyond polynomials? Rational generating functions
I For strongly polynomial sequence (Hk),
∑
k
hom(G ,Hk)tk =PG (t)
(1− t)|V (G)|+1
with polynomial PG (t) of degree at most |V (G )|.I For eventually polynomial sequence (Hk) such as (Ck),
∑
k
hom(G ,Hk)tk =PG (t)
(1− t)|V (G)|+1
with polynomial PG (t).
Page 110
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Beyond polynomials? Rational generating functions
I For quasipolynomial sequence of Turan graphs (Tk,r )
∑
k
hom(G ,Tk,r )tk =PG (t)
Q(t)|V (G)|+1
with polynomial PG (t) of degree at most |V (G )| andpolynomial Q(t) with zeros r th roots of unity.
I For sequence of hypercubes (Qk),
∑
k
hom(G ,Qk)tk =PG (t)
Q(t)|V (G)|+1
with polynomial PG (t) of degree at most |V (G )| andpolynomial Q(t) with zeros powers of 2.
Page 111
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Beyond polynomials? Rational generating functions
I For quasipolynomial sequence of Turan graphs (Tk,r )
∑
k
hom(G ,Tk,r )tk =PG (t)
Q(t)|V (G)|+1
with polynomial PG (t) of degree at most |V (G )| andpolynomial Q(t) with zeros r th roots of unity.
I For sequence of hypercubes (Qk),
∑
k
hom(G ,Qk)tk =PG (t)
Q(t)|V (G)|+1
with polynomial PG (t) of degree at most |V (G )| andpolynomial Q(t) with zeros powers of 2.
Page 112
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Beyond polynomials? Algebraic generating functions
I For sequence of odd graphs Ok = J2k−1,k−1,{0}
∑
k
hom(G ,Ok)tk
is algebraic (e.g. 12(1− 4t)−
12 when G = K1).
Page 113
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Page 114
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Four papers
P. de la Harpe and F. Jaeger, Chromatic invariants for finite graphs:theme and polynomial variations, Lin. Algebra Appl. 226–228(1995), 687–722
Defining graphs invariants from counting graph homomorphisms.Examples. Basic constructions.
D. Garijo, A. Goodall, J. Nesetril, Polynomial graph invariants fromhomomorphism numbers. Discrete Math., 339 (2016), no. 4,1315–1328. Early version at arXiv: 1308.3999 [math.CO]
Further examples. New construction using rooted treerepresentations of graphs (e.g. cotrees).
Page 115
Counting graph homomorphismsSequences giving graph polynomials
Cycle matroid invariantsOpen problems
Four papers
A. Goodall, J. Nesetril, P. Ossona de Mendez, Strongly polynomialsequences as interpretation of trivial structures. J. Appl. Logic, toappear. Also at arXiv:1405.2449 [math.CO].
General relational structures: counting satisfying assignments forquantifier-free formulas. Building new polynomial invariants byinterpretation of ”trivial” sequences of marked tournaments.
A.J. Goodall, G. Regts and L. Vena Cros, Matroid invariants andcounting graph homomorphisms. Linear Algebra Appl. 494 (2016),263–273. Preprint: arXiv:1512.01507 [math.CO]
Cycle matroid invariants from counitng graph homomorphisms.