On Two Coloring Of Planar Graphs

All graphs considered in this thesis finite and simple. For any graph G, we denote its vertex set, edge set and face set by V(G), E(G) and F(G).The colouring c is proper if no two adjacent edges have the same colour for graph G. A proper edge colouring c is called an acyclic edge coloring if there are no bichromatic cycles in the graph G. The acyclic edge chromatic nunmber is the least number of colors in an acyclic edge coloring of G. The acyclic edge k-coloring of a graph G is that there exists an acyclic edge coloring using k colors. In 2001, Alon et al gave the following conjecture(AECC for short).Conjecture 1. For any graphs G, then the acyclic edge chromatic nunmber is less than or equal maximum degree plus 2.A total coloring of a graph G is a mapping c which is form the union of the vertex set and edge set to color set C such that coloring different for every pair of adjacent or incident elements. The total chromatic number of G is the least number of colors in a total coloring of G. Behzad and Vizing gave the following conjecture (TCC for short).Conjecture 2. For any graphs G, then the total chromatic nunmber is greater than or equal maximum degree plus 1 and less than or equal maximum degree plus 2 .The thesis investigate two coloring of a class of planar graphs G, the acyclic edge coloring and the total coloring. The concrete contents and results are as follows:Firstly, the thesis summarize the current reseach on two coloring of graphs and related basic concepts, the acyclic edge coloring and total coloring.Secondly, the thesis give an upper bound on the acyclic edge chromatic number for planar graphs without 5-cycles, planar graphs with girth at least four, planar graphs without adjacents, planar graphs without 4-cycles, planar graphs not containing cycles of length 4 and 5 and planar graphs not containing any cycle of length 4,6 and 8, respectively.Finally, the thesis give planar graphs with maximum degree six contain neither 3-face with 5-vertex nor (4,6,6)-face, TCC holds.