A strengthening on consecutive odd cycles in graphs of given minimum degree

TL;DR

This paper proves a conjecture that 2-connected non-bipartite graphs with a certain minimum degree contain a specific number of consecutive odd cycles, extending previous results and confirming the conjecture for all natural numbers.

Contribution

The paper confirms Liu and Ma's conjecture for all natural numbers and improves existing results on cycles of consecutive lengths in graphs.

Findings

Confirmed the conjecture for all k in natural numbers

Established the existence of specific consecutive odd cycles in graphs

Improved bounds on cycles of consecutive lengths

Abstract

Liu and Ma [J. Combin. Theory Ser. B, 2018] conjectured that every $2$-connected non-bipartite graph with minimum degree at least $k+1$ contains $\lceil k/2\rceil $ cycles with consecutive odd lengths. In particular, they showed that this conjecture holds when $k$ is even. In this paper, we confirm this conjecture for any $k\in \mathbb N$. Moreover, we also improve some previous results about cycles of consecutive lengths.