A sufficient condition for planar graphs with girth 5 to be (1, 7)-colorable

Abstract

A graph G is $(d_1, d_2)$-colorable if its vertices can be partitioned into subsets $V_1$ and $V_2$ such that in $G[V_1]$ every vertex has degree at most $d_1$ and in $G[V_2]$ every vertex has degree at most $d_2$. Let $\mathcal {G}_5$ denote the family of planar graphs with minimum cycle length at least 5. It is known that every graph in $\mathcal {G}_5$ is $(d_1, d_2)$-colorable, where $(d_1, d_2)\in \{(2,6), (3,5),(4,4)\}$. We still do not know even if there is a finite positive d such that every graph in $\mathcal {G}_5$ is (1, d)-colorable. In this paper, we prove that every graph in $\mathcal {G}_5$ without adjacent 5-cycles is (1, 7)-colorable. This is a partial positive answer to a problem proposed by Choi and Raspaud that is every graph in $\mathcal {G}_5\;(1, 7)$-colorable?.