Sub-quorum colorings of some infinite families of caterpillars

Abstract

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.