A relaxation of the strong Bordeaux Conjecture

TL;DR

This paper proves that graphs in a specific family of plane graphs, avoiding certain cycles, are colorable with a relaxed degree constraint, advancing previous results and partially confirming a conjecture.

Contribution

The paper demonstrates that graphs in the family are (1,1,0)-colorable, relaxing the original conjecture's (0,0,0)-colorability requirement.

Findings

Graphs in the family are (1,1,0)-colorable.

Improves previous results by Xu and Liu-Li-Yu.

Partially confirms 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 $V_1,V_2, \ldots, V_k$, such that the subgraph $G[V_i]$, induced by $V_i$, has maximum degree at most $c_i$ for $i=1, 2, \ldots, k$. 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+).