加权图着色
Weighted Graph Colorings
摘要：
我们研究了两个加权图着色问题，其中一个问题是给图的顶点分配$q$颜色，使得相邻的顶点具有不同的颜色，顶点加权$w$对给定的颜色是不利的还是有利的.我们展示了一个加权的色多项式$Ph(G,q,w)$与这个问题，广义的色多项式$P(G,q)$。这个多项式的一般性质的证明，并说明计算各种家庭的图。我们表明，加权色多项式是能够区分某些图形，产生相同的色多项式。给出了具有周期纵向边界条件的格条图$G$的$Ph(G,q,w)$的一般结构公式.分析了$Ph(G,q,w)$在$q$和$w$平面上的零点及其在$G$无穷多个顶点极限下的累积集.最后，还讨论了一些相关的赋权图着色问题。
英文摘要：
We study two weighted graph coloring problems, in which one assigns $q$ colors to the vertices of a graph such that adjacent vertices have different colors, with a vertex weighting $w$ that either disfavors or favors a given color. We exhibit a weighted chromatic polynomial $Ph(G,q,w)$ associated with this problem that generalizes the chromatic polynomial $P(G,q)$. General properties of this polynomial are proved, and illustrative calculations for various families of graphs are presented. We show that the weighted chromatic polynomial is able to distinguish between certain graphs that yield the same chromatic polynomial. We give a general structural formula for $Ph(G,q,w)$ for lattice strip graphs $G$ with periodic longitudinal boundary conditions. The zeros of $Ph(G,q,w)$ in the $q$ and $w$ planes and their accumulation sets in the limit of infinitely many vertices of $G$ are analyzed. Finally, some related weighted graph coloring problems are mentioned.
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