Every Planar Graph Without Pairwise Adjacent 3-, 4-, and 5-Cycle is DP-4-Colorable
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DP-coloring is a generalization of list coloring in simple graphs. Many results in list coloring can be generalized in those of DP-coloring. Kim and Ozeki showed that every planar graph without k-cycles where \(k=3,4,5,\) or 6 is DP-4-colorable. Recently, Kim and Yu extended the result on 3- and 4-cycles by showing that every planar graph without triangles adjacent to 4-cycles are DP-4-colorable. Xu and Wu showed that every planar graph without 5-cycles adjacent simultaneously to 3-cycles and 4-cycles is 4-choosable. In this paper, we extend the results on 3-, 4-, and 5-cycles as follows. Let G be a planar graph without pairwise adjacent 3-, 4-, and 5-cycle. We prove that each precoloring of a 3-cycle of G can be extended to a DP-4-coloring of G. As a consequence, each planar graph without pairwise adjacent 3-, 4-, and 5-cycle is DP-4-colorable.
KeywordsDP-coloring List coloring Planar graph Cycle
We would like to thank anonymous referees for comments which are helpful for improvement in this paper. The first author is supported by Development and Promotion of Science and Technology talents project (DPST). The second author is supported by the Commission on Higher Education and the Thailand Research Fund under Grant RSA6180049.
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