发表机构
National Higher School of Advanced Technologies; University of Blida 1（国家高等技术学院; 布利达大学一）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究图的亚法定人数染色数，针对完全$n$元毛毛虫和最小脊柱顶点度为三的完全毛毛虫等无限族，确定了其精确值。
AI 中文摘要
图$G$的顶点集$V$的一个划分$\pi=\{V_{1},V_{2},...,V_{k}\}$，其中$V_{i}$（$i\in\{1,...,k\}$）为$k$个颜色类，若对每个顶点$v\in V$，在$v$的闭邻域$N[v]$中至少一半的顶点与$v$同色，则称该划分为{\it 法定人数染色}。$G$的法定人数染色的最大基数称为$G$的{\it 法定人数染色数}，记为$\psi_{q}(G)$。$G$的一个{\it 亚法定人数染色}是一个满射的部分函数$f:V\rightarrow\left\{1,2,\ldots,\ell\right\}$，满足性质：对每个顶点$v\in V$，若$f(v)$有定义，则在$N[v]$中具有$f$像的顶点中至少一半与$v$同色。{\it 亚法定人数染色数}$\psi_{sq}(G)$等于$G$的亚法定人数染色中最大可能的$\ell$值。本文确定了某些无限毛毛虫族的亚法定人数染色数的精确值，包括完全$n$元毛毛虫和最小脊柱顶点度为三的完全毛毛虫。
英文摘要
A partition $π=\{V_{1},V_{2},...,V_{k}\}$ of the vertex set $V$ of a graph $G$ into $k$ color classes $V_{i},$ with $i\in\{1,...,k\}$ is called a {\it quorum coloring} if for every vertex $v\in V,$ at least half of the vertices in the closed neighborhood $N[v]$ of $v$ have the same color as $v.$ The maximum cardinality of a quorum coloring of $G$ is called the {\it quorum coloring number} of $G$ and is denoted by $ψ_{q}(G).$ A {\it sub-quorum coloring} of $G$ is an onto partial function $f:V\rightarrow\left\{1,2,\ldots,\ell\right\}$ having the property that for every vertex $v\in V,$ if $f(v)$ is defined, then at least half of the vertices in $N[v]$ having an image by $f$, have the same color as $v.$ The {\it sub-quorum coloring number} $ψ_{sq}(G)$ equals the maximum value $\ell$ in a sub-quorum coloring of $G.$ In this paper, we determine the exact value of the sub-quorum coloring number for some infinite families of caterpillars including complete $n$-tuple caterpillars and complete caterpillars with minimum spine-vertex degree three.
