Injective Coloring Of Planar Graphs On The Condition Of Girth At Least 5

All graphs considered in this paper are finite simple graphs.An injective k-coloring of a graph G is an assignment of k colors to the vertices of G so that any two ver-tices with a common neighbor receive distinct colors.The injective chromatic number Xi(G),is the least k such that G is injectively k-colorable.An injective coloring is not a proper coloring.It's not hard to find that ?(G)??i(G)?Xi(G)(?i(G)-1)+ 1.Also we know Xi(G)??(G),where G ? K2.Injective coloring was introduced by Hahn et al.,who applied the injective chro-matic number to the hypercube in the theory of error-correcting codes.It turns out that the problem of L(0,1)-Labeling on graphs and the coloring of the square of G are closely interrelated with injective coloring.Today,the studying of injective chromatic number are mainly based on the girth,the maximum degree and the maximum average degree of graphs.In this paper,we mainly discuss the injective chromatic number without some short cycles.In the first part,we introduce several background knowledge and the research status of injective coloring.In the second part,we introduce a sufficient con-dition of the injective chromatic number of a graph of girth at least 5 is ? + 3.In the third part,we introduce some problems worth investigating.