Discussiones Mathematicae Graph Theory 33(2) (2013) 411-428
doi: 10.7151/dmgt.1678

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On Closed Modular Colorings of Trees

Bryan Phinezy and Ping Zhang

Department of Mathematics
Western Michigan University
Kalamazoo, MI 49008, USA


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) → ℤk be a vertex coloring where adjacent vertices may be assigned the same color. The coloring c induces another vertex coloring c ′: V(G) → ℤk defined by c ′(v) = ∑u ∈ N[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 ℤk 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. The closed modular chromatic number is investigated for trees and determined for several classes of trees. For each tree T in these classes, it is shown that mc(T) = 2 or mc(T) = 3. A closed modular k-coloring c of a tree T is called nowhere-zero if c(x) ≠ 0 for each vertex x of T. It is shown that every tree of order 3 or more has a nowhere-zero closed modular 4-coloring.

Keywords: trees, closed modular k-coloring, closed modular chromatic number

2010 Mathematics Subject Classification: 05C05, 05C15.


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Received 11 January 2012
Accepted 11 June 2012