# On subgraphs of 2k-cycle-free graphs

## Document Type

Presentation Abstract

12-4-2017

## Abstract

Kühn and Osthus showed that every bipartite 2k-cycle-free graph G contains a four-cycle-free subgraph with at least 1/(k-1) fraction of the edges of G. We give a new and simple proof of this result.

In the same paper Kühn and Osthus also showed that a 2k-cycle-free graph which is obtained by pasting together four cycles has average degree at most 16k and asked whether there exists a number d=d(k) such that every 2k-cycle-free graph which is obtained by pasting together 2l-cycles has average degree at most d if k > l \ge 3 are given integers. We answer this question negatively.

We show that for any \varepsilon>0, and any integer k \ge 2, there is a 2k-cycle-free graph G which does not contain a bipartite subgraph of girth greater than 2k with more than \left(1-\frac{1}{2^{2k-2}}\right)\frac{2}{2k-1}(1+\varepsilon) fraction of the edges of G. Győri et al. showed that if c denotes the largest constant such that every 6-cycle-free graph G contains a bipartite subgraph which is 4-cycle-free having c fraction of edges of G, then \frac{3}{8}\le c\le\frac{2}{5}. Putting k=3, our result implies that c=\frac{3}{8}.

Our proof uses the following statement, which we prove using probabilistic ideas, generalizing a theorem of Erdős: For any \varepsilon>0, and any integers a,b, k\ge2, there exists an a-uniform hypergraph H of girth greater than k which does not contain any b-colorable subhypergraph with more than \left(1-\frac{1}{b^{a-1}}\right)\left(1+\varepsilon\right) fraction of the hyperedges of H.

Joint work with Grósz and Tompkins.

Download the attached PDF to see the abstract with proper math formatting.