Extremal problems for cycles in graphs

Abstract

In this survey we consider extremal problems for cycles of prescribed lengths in graphs. The general extremal problem is cast as follows: if \(\mathcal{C}\) is a set of cycles, determine the largest number of edges \(\mathrm{ex}(n,\mathcal{C})\) in an n-vertex graph containing no cycle from \(\mathcal{C}\). The survey contains short proofs of various known theorems, including the even cycle theorem of Erdős and Bondy and Simonovits. We also give proofs of new results and conjectures of Erdős on cycles, for instance, we find new sufficient conditions for cycles of length l modulo k and for long cycles in triangle-free graphs of large chromatic number. We also review proofs of some conjectures of Erdős on the distribution of the lengths of cycles in graphs, as well as related problems on chromatic number and girth, counting graphs without short cycles, and extensions to cycles in uniform hypergraphs. Throughout the survey, we include a number of conjectures and open problems.