1 Independence densities of hypergraphs Anthony Bonato Ryerson University 2014 CMS Summer Meeting Independence densities - Anthony Bonato.

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Independence densities of hypergraphs

Anthony BonatoRyerson University

2014 CMS Summer Meeting

Independence densities - Anthony Bonato

Paths


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…• number of

independent sets = F(n+2)

- Fibonacci numberPn

Stars


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• number of independent sets = 1+2n

• independence density

= 2-n-1+½

K1,n

…

Independence density

• G order n

• i(G) = number of independent sets in G (including ∅)

– Fibonacci number of G

• id(G) = i(G) / 2n

– independence density of G


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Properties

• if G is a spanning subgraph of H, then

i(H) ≤ i(G)

• i(G U H) = i(G)i(H)

• if G is subgraph of H, then id(H) ≤ id(G)– G has an edge, then: id(G) ≤ id(K2) = 3/4

• id(G U H) = id(G)id(H)


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Infinite graphs?

• view countably a infinite graph as a limit of chains

• extend definition by continuity:

• well-defined?• possible values?


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Chains

• let G be a countably infinite graph

• a chain C in G is a set of induced subgraphs Gi such that:– for all i, Gi is an induced subgraph of Gi+1 and

• write id(G, C) =


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Existence and uniqueness

Theorem (B,Brown,Kemkes,Pralat,11)

Let G be a countably infinite graph.

1. For each chain C, id(G, C) exists.

2. For all chains C and C’ in G,

id(G, C)=id(G, C’).


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Examples

• stars: id(K1,∞) = 1/2

• one-way infinite path: id(P∞) = 0


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Bounds on id

• if G contains and infinite matching, then id(G) = 0• matching number of G, written µ(G), is the supremum of

the cardinalities of pairwise non-intersecting edges in G

Theorem (B,Brown,Kemkes,Pralat,14)

If µ(G) is finite, then:

• in particular, id(G) = 0 iff µ(G) is infinite


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Rationality

Theorem (BBKP,11)

Let G be a countable graph.

1. id(G) is rational.

2. The closure of the set

{id(G): G countable}

is a subset of the rationals.


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Aside: other densities

• many other density notions for graphs and hypergraphs:– upper density– homomorphism density– Turán density– co-degree density– cop density, …


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Question

• hereditary graph class X: closed under induced subgraphs

• egs: X = independent sets; cliques; triangle-free graphs;

perfect graphs; H-free graphs

• Xd(G) = proportion of subsets which induce a graph in X– generalizes to infinite graphs via chains

• Is Xd(G) rational?


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Hypergraphs

• hypergraph H = (V,E), E = hyperedges• independent set: does not contain a

hyperedge

• id(H) defined analogously– extend to infinite hypergraphs by continuity– well-defined


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1

2 3

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Examples

∅,{1},{2},{3},{4},{1,2},{1,3},{2,3},{1,4},{3,4},{1,3,4}

id(H) = 11/16H

Examples, cont


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…

id(H) = 7/8

Hypergraph id’s

examples:

1. graph, E = subsets of vertices containing a copy of K2

– recovers the independence density of graphs

2. graph, fix a finite graph F; E = subsets of vertices containing a copy of F– F-free density (generalizes (1)).

3. relational structure (graphs, digraphs, orders, etc); F a set of finite structures; E = subsets of vertices containing a member of F – F-free density of a structure (generalizes (2))


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Bounds on id

• matching number of hypergraph H, written µ(H), is the supremum of the cardinalities of pairwise non-intersecting hyperedges in H

Theorem (B,Brown,Mitsche,Pralat,14)

Let H be a hypergraph whose hyperedges have cardinality bounded by k > 0. If µ(H) is finite, then:

• sharp if k = 1,2; not sure lower bound is sharp if k > 2


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Rationality

• rank k hypergraph: hyperedges bounded in cardinality by k > 0– finite rank: rank k for some k

Theorem (BBMP,14): If H has finite rank,

then id(H) is rational.


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Sketch of proof

• notation: for finite disjoint sets of vertices A and B

idA,B(H) = density of independent sets containing A and not B

• analogous properties to id(H) = id ,∅ ∅(H)


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Properties of idA,B(H)

1. For a vertex x outside A U B:

.

2. For a set W outside A U B:


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Out-sets

• for a given A, B, and any hyperedge S such that S∩B = ∅, the set S \ A is the out-set of S relative to A and B – example: A B

S• notation: idr

A,B(H) denotes that every out-set has cardinality at most r

• note that: idk,∅ ∅(H) = id(H)


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Claims

1. If A is not independent, then idrA,B(H) = 0.

2. If A is independent and there is an infinite family of disjoint out-sets, then idr

A,B(H) = 0.

3. If A is independent, then id0A,B(H)=2-(|A|+|B|).

4. Suppose that A is independent and there is no infinite family of disjoint out-sets. If O1, O2,…, Os is a maximal family of disjoint out-sets, then for all r > 0,

where W is the union of the Oi.


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Final steps…

• start with idk,∅ ∅(H)

• by (2), assume wlog there are finitely many out-sets• as A and B are empty, the out-sets are disjoint

hyperedges with union W• by (4):

• apply induction using (1)-(3); each term is rational, or a finite sum of terms where we can apply (1)-(3)

• process ends after k steps


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Unbounded rank


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…

H

Theorem (BBMP,14) 1. The independence density of H is

.

2. The value is S is irrational.

• proof of (2) uses Euler’s Pentagonal Number Theorem

Any real number

• case of finite, but unbounded hyperedges• Hunb = {x: there is a countable hypergraph

H with id(H) = x}

Theorem (BBMP,14) Hunb = [0,1].

• contrasts with rank k case, where there exist gaps such as (1-1/2k,1)


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Independence polynomials

• H finite, independence polynomial of H wrt x > 0

i(H,x) = – for eg: id(H) = i(H,1)/2n

• example– i(Pn,x) = i(Pn-1,x) + xi(Pn-2,x),

i(P1,x)=1+x, (P2,x)=1+2x


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Independence densities at x

• (,x): defined in natural way for fixed x ≥ 0– may depend on chain if x ≠ 1– (,1) = id(H)

• examples: – (,x) = 0 for all x

• generalizes to chains with βn = o(n)

– (H,x) = 0 if x < 1:• bounded above by ((1+x)/2)n =o(1)


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Examples, continued

• +

• with chain (Pn: n ≥ 1), we derive that:

(,x) =


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Examples, continued

• for each r > 1, can choose chain C such that

) =

• r is a jumping point


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Future directions

• classify gaps among densities for given hypergraphs

• rationality of closure of set of id’s for rank k hypergraphs

• which hypergraphs have jumping points, and what are their values?


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General densities

• d: hypergraph function satisfying:– multiplicative on disjoint unions– monotone increasing on subgraphs

• d(H) well-defined for infinite hypergraphs

• properties of d(H)? for eg, when rational?


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1 Independence densities of hypergraphs Anthony Bonato Ryerson University 2014 CMS Summer Meeting Independence densities - Anthony Bonato.

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