Odd and even cycle lengths, minimum degree and chromatic number in graphs

TL;DR

This paper establishes new bounds relating cycle lengths, minimum degree, and chromatic number in graphs, improving previous results for both odd and even cycles, with applications to graph coloring and structure analysis.

Contribution

The paper provides novel bounds connecting cycle length sets, minimum degree, and chromatic number, extending and improving prior theorems for 2-connected graphs.

Findings

If $ ext{min degree} ext{ } ext{≥} 2k$, then $|L_o(G)| ext{ } ext{≥} k$.

For 2-connected graphs, bounds on $|L_o(G)|$ and $|L_e(G)|$ relate to clique minors or chromatic number.

Improved bounds on $ ext{chromatic number}$ based on cycle length sets and minimum degree.

Abstract

In this paper, we prove similar results for odd and even cycle lengths. Let $L_o(G)$ denote the set of odd cycle lengths of $G$ and $\ell_o(G)$ denote the longest odd cycle length. In 1992, Gy\'arf\'as proved that $\chi(G)\leq 2|L_o(G)|+2$, and if $w(G)\leq 2|L_o(G)|+1$, then $\chi(G)\leq 2|L_o(G)|+1$. We first prove that if $G$ is a 2-connected non-bipartite graph with $\delta(G)\geq 2k$, then $|L_o(G)|\geq k$. Moreover, if $|L_o(G)|=k$, then $2|L_o(G)|+1=\ell_o(G)$, and either $K_{2k+1}\subseteq G$ or $\chi(G)\leq 2k$. Applying this result, we prove that if $w(G)\leq 2|L_o(G)|$, then $\chi(G)\leq 2|L_o(G)|$ for $|L_o(G)|\geq 2$, improving the result of Gy\'arf\'as. We also construct a class of graphs with $w(G)=2|L_o(G)|-1$ but $\chi(G)=2|L_o(G)|$ for every $|L_o(G)|\geq 2$. Using our result, we give a short proof of a similar result of $\chi(G)$ and $\ell_o(G)$ proved by Kenkre and Vishwanathan. Our second part is about even cycle lengths. Let $L_e(G)$ denote the set of even cycle lengths of $G$ and $\ell_e(G)$ denote the longest even cycle length. In 2004, Mih\'ok and Schiermeyer proved that $\chi(G)\leq 2|L_e(G)|+3$, and if $w(G)\leq 2|L_e(G)|+2$, then $\chi(G)\leq 2|L_e(G)|+2$. We first prove that if $G$ is a 2-connected graph with $\delta(G)\geq 2k+1$, then $|L_e(G)|\geq k$. Moreover, if $|L_e(G)|=k$, then $2|L_e(G)|+2=\ell_e(G)$, and either $K_{2k+2}\subseteq G$ or $\chi(G)\leq 2k+1$. Applying this result, we prove that if $w(G)\leq 2|L_e(G)|+1$, then $\chi(G)\leq 2|L_e(G)|+1$ for $|L_e(G)|\geq 2$, improving the result of Mih\'ok and Schiermeyer. We also construct a class of graphs with $w(G)=2|L_e(G)|$ but $\chi(G)=2|L_e(G)|+1$ for every $|L_e(G)|\geq 2$. Our result can deduce a similar result of $\chi(G)$ and $\ell_e(G)$. The above results also improve some results of consecutive odd or even cycle lengths.