Bounded Degree Subgraphs of Dense Graphs Bounded Degree Subgraphs of Dense Graphs Julia B ˝ ottcher Technische Universit˝ at M ˝ unchen 13th International Conference on Random Structures and Algorithms, May 28-June 1, 2007, Tel Aviv (joint work with Mathias Schacht & Anusch Taraz) Julia B ˝ ottcher TU M˝ unchen Tel Aviv ’07
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Bounded Degree Subgraphs of Dense Graphs
Bounded Degree Subgraphs of Dense Graphs
Julia Bottcher
Technische Universitat Munchen
13th International Conference on Random Structures and Algorithms,May 28-June 1, 2007, Tel Aviv
(joint work with Mathias Schacht & Anusch Taraz)
Julia Bottcher TU Munchen Tel Aviv ’07
Bounded Degree Subgraphs of Dense Graphs
Subgraph containment problem
Question
Given a graph H, which conditions on an n-vertex graph G = (V , E)
ensure H ⊆ G?
Julia Bottcher TU Munchen Tel Aviv ’07
Bounded Degree Subgraphs of Dense Graphs
Subgraph containment problem
Question
Given a graph H, which conditions on an n-vertex graph G = (V , E)
ensure H ⊆ G?
Our aim: every graph G = (V , E) with minimum degree δ(G) ≥??contains a given graph H.
Julia Bottcher TU Munchen Tel Aviv ’07
Bounded Degree Subgraphs of Dense Graphs
Subgraph containment problem
Question
Given a graph H, which conditions on an n-vertex graph G = (V , E)
ensure H ⊆ G?
Our aim: every graph G = (V , E) with minimum degree δ(G) ≥??contains a given graph H.
H of small/fixed size
Erdos–Stone: δ(G) ≥(
χ(H)−2χ(H)−1 + o(1)
)
n =⇒ H ⊆ G.
Julia Bottcher TU Munchen Tel Aviv ’07
Bounded Degree Subgraphs of Dense Graphs
Subgraph containment problem
Question
Given a graph H, which conditions on an n-vertex graph G = (V , E)
ensure H ⊆ G?
Our aim: every graph G = (V , E) with minimum degree δ(G) ≥??contains a given graph H.
H of small/fixed size
Erdos–Stone: δ(G) ≥(
χ(H)−2χ(H)−1 + o(1)
)
n =⇒ H ⊆ G.
This talk
H is a spanning subgraph of G.
Julia Bottcher TU Munchen Tel Aviv ’07
Bounded Degree Subgraphs of Dense Graphs
Subgraph containment problem
Question
Given a graph H, which conditions on an n-vertex graph G = (V , E)
ensure H ⊆ G?
Our aim: every graph G = (V , E) with minimum degree δ(G) ≥??contains a given graph H.
Classical example:
δ(G) ≥ 12n ⇒ Ham ⊆ G DIRAC’52
This talk
H is a spanning subgraph of G.
Julia Bottcher TU Munchen Tel Aviv ’07
Bounded Degree Subgraphs of Dense Graphs
From Small Graphs to Spanning Graphs
A graph H of constant size is forced in G, when
δ(G) ≥
(
χ(H) − 2χ(H) − 1
+ o(1)
)
n.
Julia Bottcher TU Munchen Tel Aviv ’07
Bounded Degree Subgraphs of Dense Graphs
From Small Graphs to Spanning Graphs
A graph H of constant size is forced in G, when
δ(G) ≥
(
χ(H) − 2χ(H) − 1
+ o(1)
)
n.
For spanning H we need at least δ(G) ≥ χ(H)−1χ(H)
n, because:
H6⊆ G
n3 − 1
n3 − 1
n3 + 2
Julia Bottcher TU Munchen Tel Aviv ’07
Bounded Degree Subgraphs of Dense Graphs
From Small Graphs to Spanning Graphs
A graph H of constant size is forced in G, when
δ(G) ≥
(
χ(H) − 2χ(H) − 1
+ o(1)
)
n.
For spanning H we need at least δ(G) ≥ χ(H)−1χ(H)
n, because:
H6⊆ G
n3 − 1
n3 − 1
n3 + 2
Does δ(G) ≥(
χ(H)−1χ(H)
+ o(1))
n imply H ⊆ G?
Julia Bottcher TU Munchen Tel Aviv ’07
Bounded Degree Subgraphs of Dense Graphs
Big Graphs
Does δ(G) ≥(
χ(H)−1χ(H)
+ o(1))
n imply H ⊆ G?
Julia Bottcher TU Munchen Tel Aviv ’07
Bounded Degree Subgraphs of Dense Graphs
Big Graphs
Does δ(G) ≥(
χ(H)−1χ(H)
+ o(1))
n imply H ⊆ G?
δ(G) ≥ r−1r n ⇒ ⌊n
r ⌋ disj. copies of Kr ⊆ G.HAJNAL,SZEMEREDI’69
Julia Bottcher TU Munchen Tel Aviv ’07
Bounded Degree Subgraphs of Dense Graphs
Big Graphs
Does δ(G) ≥(
χ(H)−1χ(H)
+ o(1))
n imply H ⊆ G?
δ(G) ≥ 23n ⇒
replacements⊆ G
HAJNAL,SZEMEREDI’69
Julia Bottcher TU Munchen Tel Aviv ’07
Bounded Degree Subgraphs of Dense Graphs
Big Graphs
Does δ(G) ≥(
χ(H)−1χ(H)
+ o(1))
n imply H ⊆ G?
δ(G) ≥ 23n ⇒
replacements⊆ G
HAJNAL,SZEMEREDI’69
δ(G) ≥ χ(F)−1χ(F)
n ⇒ ⌊ n|F |⌋ disj. copies of F ⊆ G
Alon-Yuster conjecture KOMLOS, SARKOZY, AND SZEMEREDI ’01
Julia Bottcher TU Munchen Tel Aviv ’07
Bounded Degree Subgraphs of Dense Graphs
Big Graphs
Does δ(G) ≥(
χ(H)−1χ(H)
+ o(1))
n imply H ⊆ G?
δ(G) ≥ 23n ⇒
replacements⊆ G
HAJNAL,SZEMEREDI’69
δ(G) ≥ χ(F)−1χ(F)
n ⇒ ⌊ n|F |⌋ disj. copies of F ⊆ G
Alon-Yuster conjecture KOMLOS, SARKOZY, AND SZEMEREDI ’01
δ(G) ≥ r−1r n ⇒ (Ham)r ⊆ G.
FAN, KIERSTEAD ’95
KOMLOS, SARKOZY, AND SZEMEREDI’98
Julia Bottcher TU Munchen Tel Aviv ’07
Bounded Degree Subgraphs of Dense Graphs
Big Graphs
Does δ(G) ≥(
χ(H)−1χ(H)
+ o(1))
n imply H ⊆ G?
δ(G) ≥ 23n ⇒
replacements⊆ G
HAJNAL,SZEMEREDI’69
δ(G) ≥ χ(F)−1χ(F)
n ⇒ ⌊ n|F |⌋ disj. copies of F ⊆ G
Alon-Yuster conjecture KOMLOS, SARKOZY, AND SZEMEREDI ’01
δ(G) ≥ 23n ⇒ Ham2 ⊆ G
Posa’s conjecture FAN, KIERSTEAD ’95
KOMLOS, SARKOZY, AND SZEMEREDI’98
Julia Bottcher TU Munchen Tel Aviv ’07
Bounded Degree Subgraphs of Dense Graphs
Big Graphs
Does δ(G) ≥(
χ(H)−1χ(H)
+ o(1))
n imply H ⊆ G?
δ(G) ≥ 23n ⇒
replacements⊆ G
HAJNAL,SZEMEREDI’69
δ(G) ≥ χ(F)−1χ(F)
n ⇒ ⌊ n|F |⌋ disj. copies of F ⊆ G
Alon-Yuster conjecture KOMLOS, SARKOZY, AND SZEMEREDI ’01
δ(G) ≥ 23n ⇒ Ham2 ⊆ G
Posa’s conjecture FAN, KIERSTEAD ’95
KOMLOS, SARKOZY, AND SZEMEREDI’98
δ(G) ≥ (12 + γ)n ⇒ every spanning tree with ∆(T ) ≤ cn
log n ⊆ G.
Tree universality KOMLOS, SARKOZY, AND SZEMEREDI’95
Julia Bottcher TU Munchen Tel Aviv ’07
Bounded Degree Subgraphs of Dense Graphs
Generalizing Conjecture
Naıve conjecture
HG
Julia Bottcher TU Munchen Tel Aviv ’07
Bounded Degree Subgraphs of Dense Graphs
Generalizing Conjecture
Naıve conjecture
H
χ(H) = k
G
δ(G) ≥ ( k−1k + γ)n
Julia Bottcher TU Munchen Tel Aviv ’07
Bounded Degree Subgraphs of Dense Graphs
Generalizing Conjecture
Naıve conjecture
H
χ(H) = k
∆(H) ≤ ∆
G
δ(G) ≥ ( k−1k + γ)n
Julia Bottcher TU Munchen Tel Aviv ’07
Bounded Degree Subgraphs of Dense Graphs
Generalizing Conjecture
Naıve conjecture
For all k , ∆ ≥ 1, and γ > 0 exists n0 s.t.
H
χ(H) = k
∆(H) ≤ ∆
G
δ(G) ≥ ( k−1k + γ)n
=⇒ G contains H.
Julia Bottcher TU Munchen Tel Aviv ’07
Bounded Degree Subgraphs of Dense Graphs
Generalizing Conjecture
Naıve conjecture
For all k , ∆ ≥ 1, and γ > 0 exists n0 s.t.
H
χ(H) = k
∆(H) ≤ ∆
G
δ(G) ≥ ( k−1k + γ)n
=⇒ G contains H.
Counterexample:
H : random bipartite graph on n2 + n
2 vertices with ∆(H) ≤ ∆.
G : two cliques of size(
12 + γ
)
n sharing 2γn vertices.
K(12 +γ)nK(1
2 +γ)n
Julia Bottcher TU Munchen Tel Aviv ’07
Bounded Degree Subgraphs of Dense Graphs
Conjecture of Bollobas and Komlos
Conjecture
For all k , ∆ ≥ 1, and γ > 0 exists n0 and β > 0 s.t.
H
χ(H) = k
∆(H) ≤ ∆
G
δ(G) ≥ ( k−1k + γ)n
=⇒ G contains H.
Julia Bottcher TU Munchen Tel Aviv ’07
Bounded Degree Subgraphs of Dense Graphs
Conjecture of Bollobas and Komlos
Theorem
For all k , ∆ ≥ 1, and γ > 0 exists n0 and β > 0 s.t.
H
χ(H) = k
∆(H) ≤ ∆
bw(H) ≤ βn
G
δ(G) ≥ ( k−1k + γ)n
=⇒ G contains H.
Bandwidth:
bw(G) ≤ b if there is a labelling of V (G) by 1, . . . , ns.t. for all {i, j} ∈ E(G) we have |i − j | ≤ b.
i j
Julia Bottcher TU Munchen Tel Aviv ’07
Bounded Degree Subgraphs of Dense Graphs
Conjecture of Bollobas and Komlos
Theorem
For all k , ∆ ≥ 1, and γ > 0 exists n0 and β > 0 s.t.
H
χ(H) = k
∆(H) ≤ ∆
bw(H) ≤ βn
G
δ(G) ≥ ( k−1k + γ)n
=⇒ G contains H.
Examples for H:
Hamiltonian cycles (bandwidth 2)
graphs of constant tree width (bandwidth O(n/ log∆ n))