The total coloring of Fm ∨ Fn

Xiangen Chen, Zhitao Hu, B. Yao, Xue Zhao
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Abstract

Results on graph coloring can be used to draw conclusions about scheduling. Graph theory is a sort of models which can be applied in various science fields such as computer science, physics, biology, chemistry, strategy etc. And graph coloring is one of the chief topics in graph research. Suppose G(V,E) is a connect graph with order at least 2, k is a positive integer and f is the mapping from V(G)∪E(G) to {1, 2, ⋯, k}. If (1) for any uv, vw ∈ E(G), u≠w, we have f(uv)≠f(vw); (2) for any uv∈E(G), u≠v, we have f(u) ≠ f(v), f(u) ≠ f(uv), f(v) ≠ f(uv), then f is called a k-total coloring of graph G(denoted by k - TC of G). The total chromatic number, denoted by χt(G), is the least number of colors in a total coloring of graph G. Suppose G and H are two simple graphs, (V(G)∪E(G))∩(V (H)∪E(H)) = θ. Let V (G⋁H) = V(G)∪V(H), E(G⋁H) = E(G)∪E(H)∪{uv|u ∈V(G), v∈V(H)}, then G⋁H is called the join-graph of G and H. The total chromatic number of the join graph of two fans with orders m + 1 and n + 1 respectively is obtained in this paper.
Fm∨Fn的总着色
图着色的结果可以用来得出关于调度的结论。图论是一种可以应用于计算机科学、物理、生物、化学、战略等各个科学领域的模型。图的着色是图研究的主要课题之一。假设G(V,E)是一个阶数至少为2的连接图,k是一个正整数,f是从V(G)∪E(G)到{1,2,⋯k}的映射。如果(1)对于任意uv vw∈E(G) u≠w,则有f(uv)≠f(vw);(2)对于任意uv∈E(G), u≠v,有f(u)≠f(v), f(v)≠f(v),则f称为图G的k-全着色(记作k- TC (G))。总色数,记作χt(G),是图G的全着色中颜色的最少个数。设G和H是两个简单图,(v (G)∪E(G)∩(v (H)∪E(H) = θ。设V(G × H) = V(G)∪V(H), E(G × H) = E(G)∪E(H)∪{uv|u∈V(G), V∈V(H)},则G × H称为G与H的连接图。本文得到了分别为m + 1阶和n + 1阶扇形的连接图的总色数。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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