Cliques in $$C_4$$ C 4 -free graphs of large minimum degree

Abstract

A graph G is called $$C_4$$ -free if it does not contain the cycle $$C_4$$ as an induced subgraph. Hubenko, Solymosi and the first author proved (answering a question of Erdős) a peculiar property of $$C_4$$ -free graphs: $$C_4$$ -free graphs with n vertices and average degree at least cn contain a complete subgraph (clique) of size at least $$c'n$$ (with $$c'= 0.1c^2$$ ). We prove here better bounds $$\big ({c^2n\over 2+c}$$ in general and $$(c-1/3)n$$ when $$ c \le 0.733\big )$$ from the stronger assumption that the $$C_4$$ -free graphs have minimum degree at least cn. Our main result is a theorem for regular graphs, conjectured in the paper mentioned above: 2k-regular $$C_4$$ -free graphs on $$4k+1$$ vertices contain a clique of size $$k+1$$ . This is the best possible as shown by the kth power of the cycle $$C_{4k+1}$$ .