Injective Coloring Of Plane Graphs

All graphs considered in this paper are finite simple graphs. An injective-k color-ing of a graph G is a mapping c:V(G)â†'{1,2,…, k}, such that c(x)≠c(y) for each x, y∈V(G), whenever x, y have a common neighbor in G. If G has an injective-k coloring, then we call that G is injective-k colorable. The injective chromatic number of G, denoted by Xi(G), is the least integer k such that G has an injective k-coloring. Assign each vertex v∈V(G) a coloring set L(v), then L={L(v)|v€V(G)} is said to be a color list of G. Let L be a color list of G, if G has an injective coloring c such that c(v)€L/(v),(?)v∈V(G), then we call c is an injective L-coloring of G. If for any color list L, such that|L(v)|> k, G has an injective L-coloring, then G is said to be injective k-choosable. We use xil(G)=rain{k|G is injective k-choosable} to denote the injective chromatic number of G.This master thesis, focuses on injective coloring on the condition of without short cycles for planar graphs. We improve several previous results. In the first chapter, we give a brief survey of injective coloring and introduce some definitions. Then, we mainly discuss the list injective coloring of plane graphs with girth at least5or6, we give a smaller upper bound for list injective chromatic number of plane graphs containing no3-,4-,6-cycles at last.