Cycle lengths in graphs of given minimum degree

TL;DR

This paper investigates the existence of cycles with specific length properties in 2-connected graphs with given minimum degree, establishing conditions under which such cycles must exist or the graph's structure is exceptional.

Contribution

It provides new results on cycle length distributions in graphs of given minimum degree, including stability versions and maximum edge counts for certain cycle length constraints.

Findings

Graphs contain admissible cycles unless they are complete or bipartite.

Existence of cycles of all even lengths modulo k in certain graphs.

Maximum edges in graphs avoiding cycles of length 0 modulo k for odd k.

Abstract

In a graph, $k$ cycles are {\em admissible} if their lengths form an arithmetic progression with common difference one or two. Let $G$ be a 2-connected graph with minimum degree at least $k\geqslant 4$. We prove that \begin{itemize} \item [(1)] $G$ contains $k$ admissible cycles, unless $G\cong K_{k+1}$ or $K_{k,n-k}$; \item [(2)] $G$ contains cycles of lengths $\ell$ modulo $k$ for all even $\ell$, unless $G\cong K_{k+1}$ or $K_{k,n-k}$; \item [(3)] $G$ contains cycles of lengths $\ell$ modulo $k$ for all $\ell$, unless $G\cong K_{k+1}$ or $G$ is bipartite. \end{itemize} In addition, we show that if $k$ is even and $G$ is 2-connected with minimum degree at least $k-1$ and order at least $k+2$, then $G$ contains cycles of lengths $\ell$ modulo $k$ for all even $\ell$. These findings provide a stability analysis of the main results on cycle lengths in graphs of given minimum degree in [J. Gao, Q. Huo, C. Liu, J. Ma, A unified proof of conjectures on cycle lengths in graphs, International Mathematics Research Notices 2022 (10) (2022) 7615--7653]. As a corollary, we determine the maximum number of edges in a graph that does not contain a cycle of length 0 modulo $k$ for all odd $k$.