A Strengthening on Odd Cycles in Graphs of Given Chromatic Number
Jun Gao, Qingyi Huo, Jie Ma · SIAM Journal on Discrete Mathematics · 2021
Resolving a conjecture of Bollobás and Erdös, Gyárfás proved that every graph $G$ of chromatic number $k+1\geq 3$ contains cycles of $\lfloor\frac{k}{2}\rfloor$ distinct odd lengths. We strengthen this prominent result by showing that such $G$ contains cycles of $\lfloor\frac{k}{2}\rfloor$ consecutive odd lengths. Along the way, combining extremal and structural tools, we prove a stronger statement that every graph of chromatic number $k+1\geq 7$ contains $k$ cycles of consecutive lengths, except that some block is $K_{k+1}$. As corollaries, this confirms a conjecture of Verstraëte and answers a question of Moore and West when $k\geq6$.