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On subgraphs of C2k-free graphs and a problem of Kühn and Osthus

Part of: Graph theory

Published online by Cambridge University Press:  04 February 2020

Dániel Grósz ,
Abhishek Methuku  and
Casey Tompkins
Dániel Grósz*
Affiliation:
Department of Mathematics, University of Pisa, 56127Pisa, Italy
Abhishek Methuku
Affiliation:
École Polytechnique Fédérale de Lausanne, Switzerland, and Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, Republic of Korea
Casey Tompkins*
Affiliation:
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, 1053 Hungary Karlsruhe Institute of Technology, 76131Karlsruhe, Germany, and Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, Republic of Korea.
*
*Corresponding author. Email: ctompkins496@gmail.com
*Corresponding author. Email: ctompkins496@gmail.com
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Abstract

Let c denote the largest constant such that every C6-free graph G contains a bipartite and C4-free subgraph having a fraction c of edges of G. Győri, Kensell and Tompkins showed that 3/8 ⩽ c ⩽ 2/5. We prove that c = 38. More generally, we show that for any ε > 0, and any integer k ⩾ 2, there is a C2k-free graph $G'$ which does not contain a bipartite subgraph of girth greater than 2k with more than a fraction

$$\Bigl(1-\frac{1}{2^{2k-2}}\Bigr)\frac{2}{2k-1}(1+\varepsilon)$$
of the edges of $G'$ . There also exists a C2k-free graph $G''$ which does not contain a bipartite and C4-free subgraph with more than a fraction
$$\Bigl(1-\frac{1}{2^{k-1}}\Bigr)\frac{1}{k-1}(1+\varepsilon)$$
of the edges of $G''$ .

One of our proofs uses the following statement, which we prove using probabilistic ideas, generalizing a theorem of Erdős. For any ε > 0, and any integers a, b, k ⩾ 2, there exists an a-uniform hypergraph H of girth greater than k which does not contain any b-colourable subhypergraph with more than a fraction

$$\Bigl(1-\frac{1}{b^{a-1}}\Bigr)(1+\varepsilon)$$
of the hyperedges of H. We also prove further generalizations of this theorem.

In addition, we give a new and very short proof of a result of Kühn and Osthus, which states that every bipartite C2k-free graph G contains a C4-free subgraph with at least a fraction 1/(k−1) of the edges of G. We also answer a question of Kühn and Osthus about C2k-free graphs obtained by pasting together C2l’s (with k >l ⩾ 3).

MSC classification

Primary: 05C15: Coloring of graphs and hypergraphs
Secondary: 05C35: Extremal problems
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 29 , Issue 3 , May 2020 , pp. 436 - 454
Copyright
© Cambridge University Press 2020

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