Cycles of consecutive lengths in $3$-connected graphs

TL;DR

This paper proves a conjecture that every 3-connected nonbipartite graph with certain degree and size conditions contains cycles of consecutive lengths, extending previous results for all k ≥ 4.

Contribution

The paper confirms the conjecture that such graphs contain k cycles of consecutive lengths for all k ≥ 4, using new proof techniques.

Findings

Confirmed the conjecture for k=4 and k=5.

Extended known results to all k ≥ 4.

Used ideas from Gao, Huo, Liu, and Ma in the proofs.

Abstract

Recently Lin, Wang and Zhou have proved that every $3$-connected nonbipartite graph of minimum degree at least $k$ with $k\ge 6$ and order at least $k+2$ contains $k$ cycles of consecutive lengths. They also conjecture that this result is true for $k=4, 5.$ We prove this conjecture. Our proofs use many ideas of Gao, Huo, Liu and Ma.