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

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) k be a vertex coloring where adjacent vertices may be assigned the same color. The coloring c induces another vertex coloring c : V (G) k defined by c(v) = uN[v]c(u) for each v V (G), where N[v] is the closed neighborhood of v. Then c is called a closed modular k-coloring if c(u)c(v) in 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 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 mc¯(T) = 2 for each even rooted tree and mc¯(T) 3 if T is an odd rooted tree having no vertex with exactly one child. Exact values mc¯(T) are determined for several classes of odd rooted trees T.

rooted trees, closed modular colorings, closed modular $k$-coloring, closed modular chromatic number
Mathematical Subject Classification 2010
Primary: 05C15
Secondary: 05C05
Received: 1 February 2012
Accepted: 9 August 2012
Published: 23 June 2013

Communicated by Ann Trenk
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