A New Result On AVDT Chromatic Number Of Planar Graphs

Let ? be a total coloring of G,and let u be a vertex of G.We use C?(u)to denote the set of colors received by u or the edges incident with u under ?,and call C?(u)the set of colors of u with respect to ?.An adjacent vertex distinguishing total coloring(an AVD total coloring for short)of a graph G is a proper total coloring of G such that any pair of adjacent vertices have distinct sets of colors.The adjacent vertex distinguishing total chromatic number ?a"(G)is the smallest integer k such that G has an AVD total coloring with k colors.Zhang et al.([13])proposed the following conjecture:For any graph G with at least two vertices,?a"(G)? ?(G)+ 3.In[5],Huang and Wang proved the following theorem:For planar graphs G with ?(G)?11,?a"(G)??(G)+ 3.Cheng et al.([5])improved the result and showed that for planar graphs G with ?(G)? 10,?a”(G)? ?(G)+ 3.In[7],Huang and Wang characterized completely the AVD total chromatic num-ber of planar graphs G with ?(G)? 14,they proved the following theorem:If G is a planar graph with ?(G)? 13,then ?a"(G)? ?(G)+ 2;If G is a planar graph with?(G)>14,then ?a"(G)= ?(G)+ 2 if and only if G contains two adjacent vertices of degree ?(G).In this thesis,we will improve the bound of Huang and Wang.In pre-cise,we have the following conclusion:If G is a planar graph with ?(G)? 12,then?a"(G)? ?(G)+ 2;If G is a planar graph with ?(G)>13,then ?a"(G)=?(G)+ 2 if and only if G contains two adjacent vertices of degree ?(G).