On Equitable Total-Coloring Of Graphs

A total-coloring is said to be equitable if‖T_i|－|T_j‖≤1, where|T_i| is the elements (vertices and edges) number of the color ith. The minimum number of colors required for an equitable total-coloring of a simple graph G is denoted by x_et(G).We study equitable total-coloring of simple graphs and obtain one sufficient condition for x_et(G)=△(G)＋1 and an upper bound of x_et(G) under some conditions:Theorem 1 Let G be a simple graph. If△(G)=|V(G)|－1 and|V(G)|≡1(mod2), then x_et(G)=△(G)＋1.Theorem 2 Let G be a simple graph. If△(G)≤|V(G)|－2, thenx_et(G)≤|V(G)|.According to these theorems, we obtain the equitable total chromatic number of (multi) Join-graphs satisfying some conditions (to see Chapter 3). In Chapter 4 we obtain the equitable total chromatic numbers of P_m∨S_n, P_m∨F_n,P_m∨W_n,C_m∨S_n,C_m∨F_n and C_m∨W_n for all m,n(to see from Theorem 4 to Theorem 9). In Chapter 5 we derive the equitable total chromatic numbers of Double graphs of Star, Fan and Wheel (to see from Theorem 10 to Theorem 12). Finally, we propose two conjectures on Equitable Total-Coloring.Conjecture 1 Let G be a simple graph. If G only has a vertex with maximum degree, then x_et(G)=△(G)＋1.Conjecture 2 Let G be a simple graph. If G's vertexes with maximum degree are not adjacent, then x_et(G)=△(G)＋1.