Longer cycles in vertex transitive graphs

TL;DR

This paper improves the known lower bounds on the length of the longest cycle in connected vertex transitive graphs, showing they must contain cycles of length at least approximately n^{3/5}, refining Babai's earlier bound.

Contribution

The authors modify Babai's approach to establish a stronger lower bound on cycle length in vertex transitive graphs, advancing understanding of their structural properties.

Findings

Longest cycle length is at least (1 - o(1)) n^{3/5} in such graphs

Improves previous bound of √(3n) for cycle length

Provides new insights into the structure of vertex transitive graphs

Abstract

In 1979 Babai found a clever argument to prove that every connected vertex transitive graph on $n \ge 3$ vertices contains a cycle of length at least $\sqrt{3n}$. Here we modify his approach to show that such graphs must contain a cycle of length at least $(1 - o(1))n^{3/5}$.