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arXiv · 2511.18961

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

Xiaolin Wang·Meiduo Chen·Xueping Xu
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Abstract

We present the relations between clique number and chromatic number with given the number of odd or even or all cycle lengths. Let $L_o(G)$ be the set of odd cycle lengths of $G$ and $\ell_o(G)$ be the longest odd cycle length. Gy\'arf\'as (DM, 1992) proved a classic result: $\chi(G)\leq 2|L_o(G)|+2$, and if $w(G)\leq 2|L_o(G)|+1$, then $\chi(G)\leq 2|L_o(G)|+1$. Later, Wang (SIAM DM, 2008), Ma and Ning (SIAM DM, 2018) together determined the exact chromatic number when $|L_o(G)|=2$. We further prove that if $w(G)\leq 2|L_o(G)|$ for any $|L_o(G)|\geq 2$, then $\chi(G)\leq 2|L_o(G)|$. 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 the relation between $w(G)$ and $\chi(G)$ with given $\ell_o(G)$ proved by Kenkre and Vishwanathan (JGT, 2006). Let $L_e(G)$ be the set of even cycle lengths of $G$ and $\ell_e(G)$ be the longest even cycle length. Mih\'ok and Schiermeyer (DM, 2004) 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 further prove that if $w(G)\leq 2|L_e(G)|+1$ and $|L_e(G)|\geq 3$, then $\chi(G)\leq 2|L_e(G)|+1$. 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 the relation between $w(G)$ and $\chi(G)$ with given $\ell_e(G)$. Combining all the above results, we deduce the similar relations between $w(G)$ and $\chi(G)$ with given the number of cycle lengths or the longest cycle length.

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Xiaolin Wang, Meiduo Chen, Xueping Xu. 2025-11-24. Odd and even cycle lengths, minimum degree and chromatic number in graphs. https://arxiv.org/abs/2511.18961

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