Forbidding induced even cycles in a graph: typical structure and counting

TL;DR

This paper characterizes the typical structure of graphs avoiding induced even cycles of length at least 12, confirming a conjecture and revealing richer structures than previously known, with implications for related cycle-avoidance problems.

Contribution

It determines the typical structure of graphs with no induced 2k-cycle for all k≥6, confirming a conjecture and extending understanding of cycle-avoidance in graphs.

Findings

Typical structure of such graphs is richer than in related results

Verifies a conjecture of Balogh and Butterfield

Provides approximate structures for graphs without induced 8- or 10-cycles

Abstract

We determine, for all $k\geq 6$, the typical structure of graphs that do not contain an induced $2k$-cycle. This verifies a conjecture of Balogh and Butterfield. Surprisingly, the typical structure of such graphs is richer than that encountered in related results. The approach we take also yields an approximate result on the typical structure of graphs without an induced $8$-cycle or without an induced $10$-cycle.