Orthogonal polarity graphs and Sidon sets

Abstract

Determining the maximum number of edges in an $n$-vertex $C_4$-free graph is a well-studied problem that dates back to a paper of Erdős from 1938. One of the most important families of $C_4$-free graphs are the Erdős-Rényi orthogonal polarity graphs. We show that the Cayley sum graph constructed using a Bose-Chowla Sidon set is isomorphic to a large induced subgraph of the Erdős-Rényi orthogonal polarity graph. Using this isomorphism we prove that the Petersen graph is a subgraph of every sufficiently large Erdős-Rényi orthogonal polarity graph.

The authors would like to thank Jason Williford for noticing an error in the proof of Theorem 1.2 in the previous version. This error has now been corrected