Making a C6-free graph C4-free and bipartite

Abstract

We show that every C 6 -free graph G has a C 4 -free, bipartite subgraph with at least 3e(G)/8 edges. Our proof is probabilistic and uses a theorem of Füredi, Naor and Verstraëte on C 6 -free graphs.

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

“…Since any C 6 -free graph contains a bipartite subgraph with at least half as many edges, applying any of the results above shows that any C 6 -free graph G has a bipartite C 4 -free subgraph with at least 1/4 of the edges of G. Győri, Kensell and Tompkins [7] improved this factor by proving the following theorem. The complete graph K 5 (as well as a graph consisting of vertex-disjoint K 5 's) gives that c 2/5.…”

“…These results confirm special cases of the compactness conjecture of Erdős and Simonovits [4] which states that for every finite family F of graphs, there exists an F ∈ F such that ex(n, F ) = O(ex(n, F )). Since any C 6 -free graph contains a bipartite subgraph with at least half as many edges, using any of the results above it is easy to show that any C 6 -free graph G has a bipartite, C 4 -free subgraph with at least 1 4 of the edges of G. Győri, Kensell and Tompkins [7] improved this factor by showing that Theorem 1.2 (Győri, Kensell and Tompkins [7]). If c is the largest constant such that every C 6 -free graph G contains a C 4 -free and bipartite subgraph B with e(B) ≥ c • e(G), then 3 8 ≤ c ≤ 2 5 .…”