A Proof of the Chen--Raspaud Conjecture

Qi Wu, Yong Lu

Abstract

For every integer $k\ge2$, Chen and Raspaud conjectured that each graph $G$ with odd girth $\og(G)\ge2k+1$ and maximum average degree $\mad(G)<2+1/k$ has a $(2k+1:k)$-coloring. In this paper, we prove the conjecture.

Disclosure

“interests or personal relation- ships that could have appeared to influence the work reported in this paper. Data availability No data were used for the research described in this article. Declaration on the use of AI The authors used ChatGPT 5.6 Pro to assist in discussing proof strategies, checking proofs, and improving exposition. 10”
PDF page 10

Classification: Proof ideas or individual proof-step assistance
Multiplier: 8
Verified: 2026-08-23T05:16:44.575102Z

Structural counts

Pages 11 pdf

Theorems 2 source

Lemmas 7 source

Propositions 0 source

Corollaries 2 source

Definitions 0 source

Displayed equations 39 source

Bibliography entries 25 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file A_Proof_of_the_Chen__Raspaud_Conjecture.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.