List Total Coloring Of Special 1-planar Graphs

Graphs considered in this paper are simple, finite, undirected, nonempty and connected. A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, for a 1-planar graph G, we always assume that G has been drawn on the plane so that(a)every edge is crossed by at most one other edge;(b)the number of crossing edges is as small as possible.List coloring of graph G is a special coloring. The mapping L is called a total assignment for the graph G if it assigns a list L(a) of possible colors to each element a ∈ V U E. If there is a proper total coloring c such that c(α) ∈ L(a) for every element a G V U E, then we said that G is total-L-colorable. A graph G is said to be κ-total-choosable if G has a proper L-total-coloring for every total assignment L satisfying|L(x)|> k for every x E V U E. The list total chromatic number, denoted xl" (G), is the smallest integer k such that G is κ-total-choosable.Basing on the construction and properties of 1-planar graph G and its associate graph Gx, we use the discharging methods to obtain the list total chromatic number of some special 1-planar graph.In chapter 1, we introduce the definition and the background of research;In chapter 2, we prove that a 1-planar graph G without intersecting tri-angles is (△+1)-total-choosable if △> 16;In chapter 3, we prove that a 1-planar graph G without intersecting 3,4-cycles is (△+1)-total-choosable if △> 14.