图的$t$ -松弛2-距离圆染色
On $t$-relaxed 2-distant circular coloring of graphs
摘要：
令$k$为正整数。对于$\{0,1,\dots,k-1\}$中的任意两个整数$i$和$j$，令$|i-j|_k=\min\{|i-j|,k-|i-j|\}$为$i$和$j$之间的圆距离。设$t$为非负整数。假设$f$是从$V(G)$到$\{0,1,\dots,k-1\}$的映射。如果相邻顶点接收不同的整数，并且对于$G$的每个顶点$u$，$u$与$|f(u)-f(v)|_k=1$的邻居$v$的数量至多为$t$，则$f$称为$t$ -松弛2-远距离圆形$k$ -染色，或简称为$G$的$(\frac{k}{2},t)^*$ -染色。如果$G$具有$(\frac{k}{2},t)^*$ -着色，则$G$称为$(\frac{k}{2},t)^*$ -可着色。本文证明了对任意两个固定整数$k$和$t$，若$k\geq2$和$t\geq1$为$k\geq2$和$t\geq1$，则除了多项式可解的情形$k=2$和$k=3$和$t\leq3$外，判定$G$是否$(\frac{k}{2},t)^*$ -可着色是NP-完全的.对任意外平面图$G$，e证明了所有外平面图都是$(\frac{5}{2},4)^*$ -可着色的，证明了不存在固定的正整数$t$使得所有外平面图都是$(\frac{4}{2},t)^*$ -可着色的.
英文摘要：
Let $k$ be an positive integer. For any two integers $i$ and $j$ in $\{0,1,\dots,k-1\}$, let $|i-j|_k=\min\{|i-j|,k-|i-j|\}$ be the circular distance between $i$ and $j$. Let $t$ be a nonnegative integer. Suppose $f$ is a mapping from $V(G)$ to $\{0,1,\dots,k-1\}$. If adjacent vertices receive different integers, and for each vertex $u$ of $G$, the number of neighbors $v$ of $u$ with $|f(u)-f(v)|_k=1$ is at most $t$, then $f$ is called a $t$-relaxed 2-distant circular $k$-coloring, or simply a $(\frac{k}{2},t)^*$-coloring of $G$. If $G$ has a $(\frac{k}{2},t)^*$-coloring, then $G$ is called $(\frac{k}{2},t)^*$-colorable. In this paper, we prove that, for any two fixed integers $k$ and $t$ with $k\geq2$ and $t\geq1$, deciding whether $G$ is $(\frac{k}{2},t)^*$-colorable is NP-complete expect the case $k=2$ and the case $k=3$ and $t\leq3$, which are polynomially solvable. For any outerplanar graph $G$, e show that all outerplanar graphs are $(\frac{5}{2},4)^*$-colorable, we prove that there is no fixed positive integer $t$ such that all outerplanar graphs are $(\frac{4}{2},t)^*$-colorable.
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