The Total Chromatic Number Of Some Particular Planar Graphs

The colorings of the graphs is one of the main problem in studying graph theory. In general,the colorings of the graphs is divided into vertice-coloring ,edge-coloring and total coloring .et. We discussed the total colorings of planar graphs in this paper.All graphs considered in this paper are finite simple planar graphs.LetG = G(V(G),E(G),φ_G) be a graph , where V(G), E(G)denote the vertex set and edge set of G respectively. Andφ_G is an incidence function that associates each edge of G with a pair of vertices of G. If v∈V(G) ,we use d(v) stand for the degree of v.Letâ–³(G) andδ(G) denote the maximum and the minimum vertex degree ofG.We offen substituteV,E,v,andεfoTV(G),E(G),v(G),andε(G) respectively.If a graph G can be drawn in the plane so that each pair of edges intersect only at their ends,it is said to be a planar graph. A planar graph partitions the rest of the plane into a number of connected regions;the closures of these regions are called the face ofG .We denote the set of faces of a planar graph G by F. The degree of a face f in G is the number of edges of G incident with f ,each cut edges counting as twoedges.And we use d(f) or r(f) stand for the degree of f .The faces f₁ and f₂ are said to be adjacent at edge e when e is one of the commom edges of their boundaries. If G is a connected planar graph then |V| -|E| +|F| =2(Euler formula).Partitioning a set of objects into classes according to certain rules is a fundamental process in mathematics. A conceptually simple set of rules tells us for each pair of objects whether or not they are allowed in the same class. The theory of graph coloring deals with exactly this situation.Time tabling,scheduling,and sequencing problems,in their many form,are basically of this nature.A total k -coloring of G is a simultaneous coloring of the vertices and edges ofG with at most k colors. The total chromatic number x_T(G) is the smallest interger k such that G has a total k -coloring. If the total chromatic number of G isâ–³(G)+1 ,then the graph G is called class one. On the total coloring of G ,the main results known by us are the following.(1) If G is a planar graph andâ–³(G)≠6, then x_T(G)≤△(G) + 2;(2) If G is a planar graph andâ–³(G)≥9, then x_T(G) =â–³(G) + 1.We discussed the total colorings of some particular planar graphs in this paper.The paper divided into three chapters. In Chapter One,we introduce some basic conceptions of graph theory,the primary results about the theory of total coloring and several results of the paper. In Chapter Two,we discussed the particular planar graphs which 3-cycles atmost incident with two k - cycles( k = 3,4,5) ,the result is the following theorem.(1) The total coloring conjecture holds for the planar graphs which 3-cycles i ncident with at most two k- cycles(k = 3,4,5).In Chapter Three,we give some plenitude conditions for planar graphs of class one.(2) Let G be a planar graph with maximum degree 6 which 4-cycles have no common edge. If G has no 3-cycles, then X_T(G) =â–³(G) + 1.(3) Let G be a planar graph with maximum degree 7 which a 4-cycle has no common edges with k -cycle (k = 3,4). If one vertex associate with at most two3-cycles, then X_T(G) =â–³(G) + 1. (4) If G be a planar graph with maximum degree 8 which 3-cycles incident w ith at most two k- cycles(k = 3,4,5), then X_T(G) =â–³(G) + 1 .