R-hued Coloring Of Planar Graphs

A graph G is a pair G= (V, E), which contains two finite nonempty set V and E. Each element in set V is called vertex and Each element in set E of 2-element subsets of V is called edge. The set V and E are the vertex set and edge set of G. A graph G is planar if and only if it can be drawn on the plane so that each edge is not crossed by one other edge.Let k,r be integers with k>0 and r> 0, and let k={1,2,……k}.If c:V(G)â†' k, and if V (?)V(G), then define c(V’)={c(v)|v∈V’}. A (k;r)-coloring of a graph G is a mapping c:V{G)â†'k satisfying both the following.(C1) c(u)≠c(v) for every edge uv∈E(G);(C2)|c(NG(u))|≥mm{dG(v),r} for any v∈V(G).The condition (C2) is often referred to as the r-hued condition. For a fixed integer r> 0, the r-hued chromatic number of G, denoted by Xr(G), is the smallest integer k such that G has a (k, r)-coloring.In this article, we mainly study the r-hued coloring of planar graphs with girth at least 5 and 3-hued coloring of planar graphs.In the first chapter, we give the background and present situation of interrelated filed and some basic notions.In the second chapter, we mainly prove the results about r-hued coloring of planar graphs with girth at least 5:When 3≤r≤7,If G is a planar graph with g≥5, then Xr(G)≤r+11, especially, when r= 3,4,5,6, then xs{G)≤10, x4(G)≤11, X5(G)≤12, X6(G)≤13.In the third chapter, we mainly give the upper bound about 3-hued coloring of planar graphs:X3(G)≤12.In the fourth chapter, we mainly summarize the conclusions briefly and give the prospects.