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首页> 外文期刊>The Journal of Combinatorial Mathematics and Combinatorial Computing >Graphs with small upper line-distinguishing and upper harmonious chromatic numbers
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Graphs with small upper line-distinguishing and upper harmonious chromatic numbers

机译：具有较小的上界线可区分和较高的谐色数的图

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摘要

A k-line-distiriguishing coloring of a graph G = (V, E) is a partition of V into k sets V_1,..., V_k such that q() ≤ 1 for i = 1,..., k and q(V_i, V_j) ≤ 1 for 1 ≤ i < j ≤ k. If the color classes in a line-distinguishing coloring is 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. For any graph G of maximum degree Δ(G), H'(G) ≥ Δ(G) arid H(G) ≥ Δ(G)+1. We characterize connected graphs G that contain neither a triangle nor a 5-cycle for which H(G) = Δ(G) + 1. We show that a triangle-free connected graph G satisfies H'(G) = Δ(G) if and only if G is a star K_(1,Δ(G)). A partial characterization of connected graphs G for which H'(G) = Δ(G) is obtained.
机译：图G =（V，E）的k线可区分的着色是V划分为k个集合V_1，...，V_k的分区，使得对于i = 1，...，q（）≤1。 ，对于1≤i

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