Exact results for generalized extremal problems forbidding an even cycle

TL;DR

This paper provides exact extremal numbers for the maximum copies of complete bipartite graphs and cycles in large graphs that exclude certain even cycles, advancing understanding in extremal graph theory.

Contribution

It determines exact maximum counts of specific subgraphs in large, cycle-free graphs, extending extremal results to new cases and small graphs.

Findings

Maximum copies of $K_{s,s}$ in $C_{2s+2}$-free graphs for large n

Maximum cycles of length $2s$ in $C_{2s+2}$-free bipartite graphs for small s

Exact extremal numbers for these subgraph counts

Abstract

We determine the maximum number of copies of $K_{s,s}$ in a $C_{2s+2}$-free $n$-vertex graph for all integers $s \ge 2$ and sufficiently large $n$. Moreover, for $s\in\{2,3\}$ and any integer $n$ we obtain the maximum number of cycles of length $2s$ in an $n$-vertex $C_{2s+2}$-free bipartite graph.