Sparse halves in K4‐free graphs

Abstract

A conjecture of Chung and Graham states that every K 4‐free graph on n vertices contains a vertex set of size ⌊ n ∕ 2 ⌋ that spans at most n 2 ∕ 18 edges. We make the first step toward this conjecture by showing that it holds for all regular graphs.

K4$K_4$‐free graphs have sparse halves

“…We proceed with a stability result addressing extremal graphs. With some additional assumptions on the degree distribution this was obtained earlier by X. Liu and J. Ma in [13,Lemma 4.10].…”

“…In this subsection we study lower bounds on the number of edges spanned by the neighbourhood of a vertex in an extremal graph. We commence with the following variant of a result by X. Liu and J. Ma, see [13, Theorem 1.5(1)]. Lemma Let $$m$$, $$n$$, $$q$$ be positive integers such that $$q\geqslant \frac{2}{9}m^2$$ and $$n\geqslant m$$.…”

“…X. Liu and J. Ma proved in [13, Theorem 4.1] that $$e(G) > \frac{1}{4}v(G)^2$$ holds for every extremal graph $$G$$. In fact, they even obtained such a result with $$\frac{1}{4}$$ replaced by $$\frac{32}{123}$$.…”

“…The most recent contribution to this problem is due to X. Liu and J. Ma [13], who proved Theorem 1.1 under the additional assumption that $$G$$ is regular. Several parts of their argument do not depend on degree regularity and we shall utilise some of their results.…”

K_4-free graphs have sparse halves

“…The most recent contribution to this problem is due to X. Liu and J. Ma [13], who proved Theorem 1.1 under the additional assumption that G be regular. Several parts of their argument do not depend on degree regularity and we shall utilise some of their results.…”