The homomorphism threshold of ‐free graphs

Abstract

We determine the structure of {C3,C5}‐free graphs with n vertices and minimum degree larger than n/5: such graphs are homomorphic to the graph obtained from a (5k−3)‐cycle by adding all chords of length 1(mod5), for some k. This answers a question of Messuti and Schacht. We deduce that the homomorphism threshold of {C3,C5}‐free graphs is 1/5, thus answering a question of Oberkampf and Schacht.

On the structure of Dense graphs with bounded clique number

“…, C 2k+1 } of odd cycles of length at most 2k + 1. We remark that for these families the homomorphism threshold was obtained and in [4,10] it was shown that δ hom (C 2k+1 ) = 1/(2k + 1).…”

“…In particular, the case of odd cycles of length at least five seems to be an interesting open case. Very recently, in [4,10] an upper bound of the form…”

On the local density problem for graphs of given odd‐girth

“…Analogous to the relation between Conjecture and Theorem one may wonder if every ‐dense graph contains an odd cycle of length at most . Letzter and Snyder showed that a graph G on n vertices with and odd‐girth at least seven is homomorphic to , for some k . Therefore combining this result with Theorem we get the following.…”

Homomorphism Thresholds for Odd Cycles

“…Note that for k " 2 part (ii ) of Theorem 1.2 would include part (i ) and this is Łuczak's theorem [13]. For k " 3 Theorem 1.2 was obtained by Letzter and Snyder [12]. We remark that our approach substantially differs from the work of Łuczak and of Letzter and Snyder.…”

“…A first step of generalising Łuczak's result by viewing K 3 as the shortest odd cycle, was recently undertaken by Letzter and Snyder [12] by showing δ hom pC 5 q ď 1 5 and δ hom ptC 3 , C 5 uq " 1 5 .…”