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

TL;DR

This paper advances the understanding of how the diversity and extremal lengths of odd and even cycle lengths constrain a graph's chromatic number. By establishing tight relationships between |L_o(G)|, |L_e(G)|, and χ(G) under minimum degree and clique constraints, it extends Gyárfás and Mihók–Schiermeyer results and yields sharper bounds, including tight equality cases tied to complete graphs. The authors employ induction, block decomposition, and intricate cycle-chord analyses, yielding parallel odd- and even-cycle theories and tightening connections to consecutive-cycle-length results. The work also provides constructions demonstrating tightness and derives corollaries that recover or strengthen prior bounds on χ(G) in terms of cycle-length structure.

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, Gyárfás proved that $χ(G)\leq 2|L_o(G)|+2$, and if $w(G)\leq 2|L_o(G)|+1$, then $χ(G)\leq 2|L_o(G)|+1$. We first prove that if $G$ is a 2-connected non-bipartite graph with $δ(G)\geq 2k$, 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 $χ(G)\leq 2k$. Applying this result, we prove that if $w(G)\leq 2|L_o(G)|$, then $χ(G)\leq 2|L_o(G)|$ for $|L_o(G)|\geq 2$, improving the result of Gyárfás. We also construct a class of graphs with $w(G)=2|L_o(G)|-1$ but $χ(G)=2|L_o(G)|$ for every $|L_o(G)|\geq 2$. Using our result, we give a short proof of a similar result of $χ(G)$ 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, Mihók and Schiermeyer proved that $χ(G)\leq 2|L_e(G)|+3$, and if $w(G)\leq 2|L_e(G)|+2$, then $χ(G)\leq 2|L_e(G)|+2$. We first prove that if $G$ is a 2-connected graph with $δ(G)\geq 2k+1$, 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 $χ(G)\leq 2k+1$. Applying this result, we prove that if $w(G)\leq 2|L_e(G)|+1$, then $χ(G)\leq 2|L_e(G)|+1$ for $|L_e(G)|\geq 2$, improving the result of Mihók and Schiermeyer. We also construct a class of graphs with $w(G)=2|L_e(G)|$ but $χ(G)=2|L_e(G)|+1$ for every $|L_e(G)|\geq 2$. Our result can deduce a similar result of $χ(G)$ and $\ell_e(G)$. The above results also improve some results of consecutive odd or even cycle lengths.

Figures (2)

• Figure 1: $H_x$ with $s=1$ and $t=k=3$
• Figure 2: Some $(x_0,v)$-paths

Theorems & Definitions (28)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9
• Theorem 10