On closed modular colorings of rooted trees

Phinezy, Bryan; Zhang, Ping

doi:10.2140/involve.2013.6.83

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On closed modular colorings of rooted trees

Bryan Phinezy and Ping Zhang

Vol. 6 (2013), No. 1, 83–97

DOI: 10.2140/involve.2013.6.83

Abstract

Two vertices $u$ and $v$ in a nontrivial connected graph $G$ are twins if $u$ and $v$ have the same neighbors in $V (G) - {u, v}$ . 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 $c : V (G) \to ℤ_{k}$ be a vertex coloring where adjacent vertices may be assigned the same color. The coloring $c$ induces another vertex coloring $c^{'} : V (G) \to ℤ_{k}$ defined by $c^{'} (v) = \sum_{u \in N [v]} c (u)$ for each $v \in V (G)$ , where $N [v]$ is the closed neighborhood of $v$ . Then $c$ is called a closed modular $k$ -coloring if $c^{'} (u) \neq c^{'} (v)$ 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 $\bar{mc} (G)$ 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 $\bar{mc} (T) = 2$ for each even rooted tree and $\bar{mc} (T) \leq 3$ if $T$ is an odd rooted tree having no vertex with exactly one child. Exact values $\bar{mc} (T)$ are determined for several classes of odd rooted trees $T$ .

Keywords

rooted trees, closed modular colorings, closed modular $k$-coloring, closed modular chromatic number

Mathematical Subject Classification 2010

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

Vol. 6, No. 1, 2013