#### Vol. 6, No. 1, 2013

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On closed modular colorings of rooted trees

### Bryan Phinezy and Ping Zhang

Vol. 6 (2013), No. 1, 83–97
##### 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 $c:V\left(G\right)\to {ℤ}_{k}$ be a vertex coloring where adjacent vertices may be assigned the same color. The coloring $c$ induces another vertex coloring ${c}^{\prime }:V\left(G\right)\to {ℤ}_{k}$ 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 ${ℤ}_{k}$ 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 $T$.

##### Keywords
rooted trees, closed modular colorings, closed modular $k$-coloring, closed modular chromatic number
Primary: 05C15
Secondary: 05C05
##### Milestones
Received: 1 February 2012
Accepted: 9 August 2012
Published: 23 June 2013

Communicated by Ann Trenk
##### Authors
 Bryan Phinezy Department of Mathematics Western Michigan University Kalamazoo, MI 49008 United States Ping Zhang Department of Mathematics Western Michigan University 1903 W. Michigan Avenue Kalamazoo, MI 49008 United States