A total k-coloring of a graph G is a coloring of V(G) ∪ E(G) using k colors such that no two adjacent or incident elements receive the same color.The total chromatic number χ〃(G) is the smallest integer k such that...A total k-coloring of a graph G is a coloring of V(G) ∪ E(G) using k colors such that no two adjacent or incident elements receive the same color.The total chromatic number χ〃(G) is the smallest integer k such that G has a total k-coloring.In this paper,it is proved that the total chromatic number of any graph G embedded in a surface Σ of Euler characteristic χ(Σ)≥0 is Δ(G) + 1 if Δ(G)≥10,where Δ(G) denotes the maximum degree of G.展开更多
It is proved that if G is a (+1)-colorable graph, so are the graphs G×Pn and C×Cn, where Pn and Cn are respectively the path and cycle with n vertices, and the maximum edge degree of the graph. The exact ch...It is proved that if G is a (+1)-colorable graph, so are the graphs G×Pn and C×Cn, where Pn and Cn are respectively the path and cycle with n vertices, and the maximum edge degree of the graph. The exact chromatic numbers of the product graphs and are also presented. Thus the total coloring conjecture is proved to be true for many other graphs.展开更多
基金supported by the Research Foundation for Doctor Programme (Grant No.200804220001)National Natural Science Foundation of China (Grant Nos.10871119,10971121,10901097)
文摘A total k-coloring of a graph G is a coloring of V(G) ∪ E(G) using k colors such that no two adjacent or incident elements receive the same color.The total chromatic number χ〃(G) is the smallest integer k such that G has a total k-coloring.In this paper,it is proved that the total chromatic number of any graph G embedded in a surface Σ of Euler characteristic χ(Σ)≥0 is Δ(G) + 1 if Δ(G)≥10,where Δ(G) denotes the maximum degree of G.
基金Project supported by National Natural Science Foundation(No. 69882002) and "973" project (No. G1999035805)
文摘It is proved that if G is a (+1)-colorable graph, so are the graphs G×Pn and C×Cn, where Pn and Cn are respectively the path and cycle with n vertices, and the maximum edge degree of the graph. The exact chromatic numbers of the product graphs and are also presented. Thus the total coloring conjecture is proved to be true for many other graphs.
