On closed modular colorings of rooted trees

Abstract

Two vertices $u$ and $v$ in a nontrivial connected graph $G$ are twins if $u$ and $v$ have the same neighbors in $V\left(G\right)-\left\{u,v\right\}$. If $u$ and $v$ are adjacent, they are referred to as true twins, while if $u$ and $v$ are nonadjacent, they are false twins. For a positive integer $k$, let $c:V\left(G\right)\to {\mathbb{Z}}_k$ be a vertex coloring where adjacent vertices may be assigned the same color. The coloring $c$ induces another vertex coloring $c^{\prime }:V\left(G\right)\to {\mathbb{Z}}_k$ defined by $c^{\prime }\left(v\right)={\sum }_{u\in N\left[v\right]}c\left(u\right)$ for each $v\in V\left(G\right)$, where $N\left[v\right]$ is the closed neighborhood of $v$. Then $c$ is called a closed modular $k$-coloring if $c^{\prime }\left(u\right)\ne c^{\prime }\left(v\right)$ in ${\mathbb{Z}}_k$ for all pairs $u$, $v$ of adjacent vertices that are not true twins. The minimum $k$ for which $G$ has a closed modular $k$-coloring is the closed modular chromatic number $\overline{mc}\left(G\right)$ of $G$. A rooted tree $T$ of order at least 3 is even if every vertex of $T$ has an even number of children, while $T$ is odd if every vertex of $T$ has an odd number of children. It is shown that $\overline{mc}\left(T\right)=2$ for each even rooted tree and $\overline{mc}\left(T\right)\le 3$ if $T$ is an odd rooted tree having no vertex with exactly one child. Exact values $\overline{mc}\left(T\right)$ are determined for several classes of odd rooted trees $T$.