The Total Coloring Of Particular Planar Graphs

All graphs considered in this paper are finite simple graphs. Let G = G(V(G),E(G),Ψ_G) be a graph, where V(G) ≠ φ and E(G) denote the vertexset and edge set of G, Ψ_G is an incidence function that associates each edge of Gwith an unordered pair of vertices of G. A graph is said to be embeddable in the plane, or planar, if it can be drawn in the plane so that its edges intersect only at their ends. Such a drawing of a planar graph G is called a planar embedding of G . We therefore sometimes refer to a planar embedding of a planar graph as a plane graph. A plane graph G partitions the rest of the plane into a number of connected regions; the closures of these regions are called the faces of G.If v ∈ V(g), we use d(v) stand for the degree of v in G. Let △(g) andS(g) denote the maximum and the minimum vertex degree of G. A walk in G is afinite non-null sequence whose terms are alternately vetices and edges . if the vertices of walk are distinct, then the walk is called a path. A walk is closed if it has positive length and its origin and terminus are the same. A closed trail is a cycle. A cycle oflength k is called a k -cycle. The degree of a face f in G (written r(f)) is the number of edges of G incident with f,each cut edges counting as two edges. Given a graph G, a total k -coloring of G is a simultaneous coloring of the vertices and edges of G with at most k colors. The total chromatic number x_?(G) is the smallest interger k such that G has a total k -coloring. If the total chromatic...