Upper Line-Distinguishing and Upper Harmonious Chromatic Numbers of Graphs

Johannes H. Hattingh1, Michacl A. Henning 2, Elna Ungerer 3
1Departinent of Mathematics atc Statistics Georgia State University Atlanta. GA 30303 U.S.A.
2 Department of Mathematics University of Natal Private Bag X01 Pietermaritzburg. 3209 South Africa
3Department of Mathematics Rand Afrikaans University Auckland Park, 2006 South Africa

Abstract

A \(k\)-line-distinguishing coloring of a graph \(G = (V,E)\) is a partition of \(V\) into \(k\) sets \(V_1, V_2, \ldots, V_k\) such that \(q(\langle V_i \rangle) \leq 1\) for \(i = 1, \ldots, k\) and \(q(V_i . V_j) \leq 1\) for \(1 \leq i < j \leq k\). If the color classes in a line-distinguishing coloring are also independent. Then it is called a harmonious coloring. A coloring is minimal if when two color classes are combined, we no longer have a coloring of the given type. The upper harmonious chromatic number, \(H(G)\), is defined as the maximum cardinality of a minimal harmonious coloring of a graph \(G\). While the upper line-distinguishing chromatic number, \(H'(G)\), is defined as the maximum cardinality of a minimal line-distinguishing coloring of a graph \(G\). We determine \(H'(C_n)\) and \(H(C_n)\) for an even cycle \(C_n\).