Cliques in \(C_4\)-free graphs of large minimum degree
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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 \(C_{4k+1}\).
Keywords
\(C_4\)-free graphs Large cliques Regular graphsNotes
Acknowledgments
The authors are grateful to József Solymosi for conversations and to Xing Peng for his interest in the subject. Research of A. Gyárfás was supported in part by the OTKA Grant No. K104343. Research of G. N. Sárközy was supported in part by the National Science Foundation under Grant No. DMS-0968699 and by OTKA Grant No. K104343.
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