Total Coloring Of Planar Graphs

Let G =(V,E)be a finite,simple and undirected graph with the set of vertices V and the set of edges E,and C = {1,2,...,k},a set of k colors.A properκ-total-coloring of G is a mapping from V∪E into C such that any two adjacent or incident elements in V∪E receive distinct colors.If G has a properκ-total-coloring,then G is said to beκ-totally-colorable.The total chromatic number,denoted byχT(G),is the smallest non-negative integerκsuch that G isκ-totally-colorable.A total-list-assignment L of G is a set of lists of colors which assign each element x∈V∪E of G a list of colors L(x).If G has a proper total-coloringφsuch thatφ(x)∈L(x)for every x∈V∪E,then G is said to be L colorable.If G is L-colorable for every total-list-assignment L with |L(x)| =κfor every x∈V∪E,then we say that G isκ-list-totally-colorable,orκ-totally-choosable.The total choice number,or,list-chromatic number of G,denoted by ch_T(G),is the least non-negative integerκsuch that G isκ-totally-choosable.On total coloring of graphs,Vizing and Behazd independently conjec-tured that:Every simple graph G with maximum degreeΔis(Δ+ 2)-totally-colorable.This is known as Total Coloring Conjecture,in short, TCC.This dissertation mainly studies total coloring and list total coloring of planar graphs.For planar graphs G,we have:(1)IfΔ(G)= 6,and G has no adjacent triangles,then G is 8-totally-colorable;(2)IrΔ(G)= 6,and G has no 4 cycles or adjacent 5 faces,then G is 8-totally-colorable; (3)IfΔ(G)= 9,and G has no adjacent triangles,then G is 10-totally-colorable;(4)IfΔ(G)= 6,and G has no intersectant triangles,then G is 8-totally-choosable;(5)IfΔ(G)= 11,and G has no adjacent triangles,then G is 12-totally-choosable.