A Proof of the Chen--Raspaud Conjecture

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