Injective coloring of planar graphs

Authors:
Yuehua Bu;Dong Chen;André Raspaud;Weifan Wang
Affiliations:
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China;Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China;LaBRI UMR CNRS 5800, Université Bordeaux I, 33405 Talence Cedex, France;Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
Venue:
Discrete Applied Mathematics
Year:
2009

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Abstract

A coloring of a graph G is injective if its restriction to the neighborhood of any vertex is injective. The injective chromatic number@g"i(G) of a graph G is the least k such that there is an injective k-coloring. In this paper we prove that if G is a planar graph with girth g and maximum degree @D, then (1) @g"i(G)=@D if either g=20 and @D=3, or g=7 and @D=71; (2) @g"i(G)@?@D+1 if g=11; (3) @g"i(G)@?@D+2 if g=8.