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

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## 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?.

### Keywords

Planar graphs Improper coloring Discharging method## Notes

### Acknowledgments

The authors thank the referees for their careful reading and valuable suggestions. M. Chen is Research supported by NSFC (Nos. 11471293, 11271335, 11401535), ZJNSFC (No. LY14A010014). Wang is Research supported by NSFC (No. 11301035).

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