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arXiv 0909.2769math.CO
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关于图的Fall染色

On Fall Colorings of Graphs

Saeed Shaebani

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摘要：

图$G$的降$k$ -染色是$G$的真$k$ -染色，使得$G$的每个顶点都看到其闭邻域上的所有$k$颜色。我们记${\rm Fall}(G)$为所有正整数$k$的集合，其中$G$具有下降$k$ -着色。本文研究了图的字典积和图的范畴积的落染色，并回答了\cite{dun}中关于完全图的范畴积的落染色的一个问题.然后，我们研究了并图的秋天染色。然后证明了图的降$k$ -染色可以约化为图在指定集合中的真$k$ -染色。然后，我们刻画了图的Mycielskian的秋天染色。最后，证明了对每个二部图$G$，${\rm Fall}(G^{c})\subseteq \{χ(G^{c}) \}$，判定是否为${\rm Fall}(G^{c})=\{χ(G^{c}) \}$是多项式时间的.

英文摘要：

A fall $k$-coloring of a graph $G$ is a proper $k$-coloring of $G$ such that each vertex of $G$ sees all $k$ colors on its closed neighborhood. We denote ${\rm Fall}(G)$ the set of all positive integers $k$ for which $G$ has a fall $k$-coloring. In this paper, we study fall colorings of lexicographic product of graphs and categorical product of graphs and answer a question of \cite{dun} about fall colorings of categorical product of complete graphs. Then, we study fall colorings of union of graphs. Then, we prove that fall $k$-colorings of a graph can be reduced into proper $k$-colorings of graphs in a specified set. Then, we characterize fall colorings of Mycielskian of graphs. Finally, we prove that for each bipartite graph $G$, ${\rm Fall}(G^{c})\subseteq \{χ(G^{c}) \}$ and it is polynomial time to decision whether or not ${\rm Fall}(G^{c})=\{χ(G^{c}) \}$.

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