Circumference of 3-connected cubic graphs

TL;DR

This paper improves the lower bound on the circumference of 3-connected cubic graphs from approximately n^0.753 to n^0.8, advancing understanding of cycle lengths in such graphs.

Contribution

It introduces new techniques to establish a higher lower bound on the circumference of 3-connected cubic graphs, refining previous results.

Findings

Lower bound on circumference improved to Ω(n^{0.8})

Methods involve analyzing 2-connected cubic graphs and cycle construction

Distinguishes cases based on adjacency of edges in cycle analysis

Abstract

The circumference of a graph is the length of its longest cycles. Jackson established a conjecture of Bondy by showing that the circumference of a 3-connected cubic graph of order $n$ is $\Omega(n^{0.694})$. Bilinski {\it et al.} improved this lower bound to $\Omega(n^{0.753})$ by studying large Eulerian subgraphs in 3-edge-connected graphs. In this paper, we further improve this lower bound to $\Omega(n^{0.8})$. This is done by considering certain 2-connected cubic graphs, finding cycles through two given edges, and distinguishing the cases whether or not these edges are adjacent.