Equitable Total Coloring of Kn(o)del Graph WΔ,n

来源 :2014全国理论计算机科学学术年会 | 被引量 : 0次 | 上传用户:HUZHAOHUA333
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  A k-total coloring of a graph G is a map σ: V (G)∪E(G)→{1,2,...,k} such that no two adjacent or incident elements of V(G)∪E(G) receive the same color.The total chromatic number is the smallest one out of all the k such that G has a k-total coloring.A k-total coloring is equitable if ‖σ-1(i)|-|σ-1(j)‖≤1 for each pair of distinct colors i and j (1≤i,j≤k).The smallest one out of all the k for which G has an equitable total k-coloring is named equitable total chromatic number.It is known that the problem of determining the total chromatic number is NP-hard,and it remains NP-hard for Δ-regular bipartite graphs with Δ≥3.In this paper,we show that the equitable total chromatic number of Kn(o)del Graph WΔ,n is Δ+1 while Δ=3,4,5 for all even n≥2Δ except W3,10.The result determines the total chromatic numbers of Kn(o)del graphs W3,n,W4,n,and W5,n,also is relevant as an evidence that every regular graph with Δ≤5 is such that the total chromatic number is equal to the equitable total chromatic number.
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