Total Coloring Of Planar Graphs Without Adjacent Short Cycles

All graphs considered in this paper are finite,simple and undirected graphs.Fur a graph G,we denote by V(G),E(G),F(G),δ(G) and△(G) the vertex set,the edge set,the face set,the minimum(vertex) degree and the maximum(vertex) degree of G.A proper k-total-coloring of G is a mappingλ:V(G)∪E(G)→{1,2.3...k} such thatλ(x)≠λ(y) for every pair of adjacent or incident x,y∈V(G)∪E(G).If a graph G has a proper k-total-coloring,we call G is k-totally-colorable.We define the total chromatic number XT(G)=min{k}G is k-totally-colorable}.Clearly,for any graph G,XT(G)≥△(G)+1.The following conjecture was posed independently by Behzad and Vizing in 1965:Total Coloring Conjecture(TCC):For any graph G,XT(G)≤△(G)+2.For any graph,the conjecture was verified for△(G)≤5.For planar graphs,the conuecture was verified for△(G)≥7.So the only case for planar graphs that remained open is△(G)=6.For any graph G,if XT(G)=△(G)+1,then G is called to be typeⅠ;if XT(G)=△(G) + 2,then G is called to be typeⅡ.It has been proved that if G is a planar graph with△(G)≥9,then G is typeⅠ.In this paper,we consider the total coloring of planar graphs without adjacent short cycles,where short cycles means that cycles of length less or equal than 5.We use the Euler's formula and discharging method to get the following two results:If G is a planar graph with△(G)=6 and there are not adjacent short cycles in G,then XT(G)≤8;If G is a planar graph with△(G)=8 and there are not adjacent short cycles in G,then G is typeⅠ.