On the Bollobás–Eldridge Conjecture for Bipartite Graphs

Abstract

Let G be a simple graph on n vertices. A conjecture of Bollobás and Eldridge [5] asserts that if δ(G) ≥ kn−1 k+1 then G contains any n vertex graph H with ∆(H) = k. We prove a strengthened version of this conjecture for bipartite, bounded degree H, for sufficiently large n. This is the first result on this conjecture for expander graphs of arbitrary (but bounded) degree. An important tool for the proof is a new version of the Blow-up Lemma.

Extremal graph packing problems: Ore-type versus Dirac-type

“…Other interesting results on graph packing were obtained by Brandt [14], Csaba [19,20], Fan and Kierstead [28,29], Komlós [47], Komlós, Sárkőzy and Szemerédi [49,50,51], Sauer and Wang [69], Wozniak [37,67], Yap [72,73,78] and others. One can look into the 75-page survey [77] of the topic by Wozniak.…”

A Dirac-Type Result on Hamilton Cycles in Oriented Graphs

“…In the next section we introduce some basic definitions and describe the extremal example which shows that Theorem 1 (and thus also Theorems 3 and 4) is essentially best possible. Our proof of Theorem 3 relies on the Regularity lemma for digraphs and on a variant (due to Csaba [6]) of the Blow-up lemma. These and other tools are introduced in Section 3, where we also give an overview of the proof.…”

On embedding well-separable graphs

“…By a standard application of the probabilistic method one can prove that for a given r if d is large enough (d = constant · r is sufficient), then H d / ⊂ G. Since H d is bipartite for every d, this proves, that the critical parameter for embedding expanders cannot be the chromatic number. (Although, the chromatic number still has a role, see [5].) One may think, that the main reason of this fact is that H d is an expander graph with large expansion rate.…”