A relaxation of the strong Bordeaux Conjecture

Abstract

Let $c_1,c_2,\cdots ,c_k$ be $k$ non-negative integers. A graph $G$ is $(c_1,c_2,\cdots ,c_k)$-colorable if the vertex set can be partitioned into $k$ sets $V_1,V_2,\dots ,V_k$, such that the subgraph $G[V_i]$, induced by $V_i$, has maximum degree at most $c_i$ for $i=1,2,\dots ,k$. Let $\mathcal{F}$ denote the family of plane graphs with neither adjacent 3-cycles nor $5$-cycle. Borodin and Raspaud (2003) conjectured that each graph in $\mathcal{F}$ is $(0,0,0)$-colorable. In this paper, we prove that each graph in $\mathcal{F}$ is $(1,1,0)$-colorable, which improves the results by Xu (2009) and Liu-Li-Yu (2014+).

14 pages