Coloring graphs with two odd cycle lengths
Jie Ma, Bo Ning

TL;DR
This paper determines the chromatic number of graphs with exactly two specified odd cycle lengths, providing complete solutions for certain cases and improving classical bounds on graph coloring.
Contribution
The paper completely characterizes the chromatic number for graphs with two specific odd cycle lengths, extending previous results and refining classical bounds.
Findings
For $L(G)= ext{\{3,3+2l\}}$, $oxed{ ext{chromatic number}= ext{max}igrace{3, ext{clique number}}ig}$.
For $L(G)= ext{\{k,k+2l\}}$, $oxed{ ext{chromatic number}=3}$.
Provides a complete solution to the problem of coloring graphs with two odd cycle lengths.
Abstract
In this paper we determine the chromatic number of graphs with two odd cycle lengths. Let be a graph and be the set of all odd cycle lengths of . We prove that: (1) If , where , then ; (2) If , where and , then . These, together with the case solved in \cite{W}, give a complete solution to the general problem addressed in \cite{W,CS,KRS}. Our results also improve a classical theorem of Gy\'{a}rf\'{a}s which asserts that for any graph .
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
