741 research outputs found

    The harmonious chromatic number of almost all trees

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    Locally identifying coloring in bounded expansion classes of graphs

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    A proper vertex coloring of a graph is said to be locally identifying if the sets of colors in the closed neighborhood of any two adjacent non-twin vertices are distinct. The lid-chromatic number of a graph is the minimum number of colors used by a locally identifying vertex-coloring. In this paper, we prove that for any graph class of bounded expansion, the lid-chromatic number is bounded. Classes of bounded expansion include minor closed classes of graphs. For these latter classes, we give an alternative proof to show that the lid-chromatic number is bounded. This leads to an explicit upper bound for the lid-chromatic number of planar graphs. This answers in a positive way a question of Esperet et al [L. Esperet, S. Gravier, M. Montassier, P. Ochem and A. Parreau. Locally identifying coloring of graphs. Electronic Journal of Combinatorics, 19(2), 2012.]

    Harmonious Coloring of Trees with Large Maximum Degree

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    A harmonious coloring of GG is a proper vertex coloring of GG such that every pair of colors appears on at most one pair of adjacent vertices. The harmonious chromatic number of GG, h(G)h(G), is the minimum number of colors needed for a harmonious coloring of GG. We show that if TT is a forest of order nn with maximum degree Δ(T)≥n+23\Delta(T)\geq \frac{n+2}{3}, then h(T)= \Delta(T)+2, & if $T$ has non-adjacent vertices of degree $\Delta(T)$; \Delta(T)+1, & otherwise. Moreover, the proof yields a polynomial-time algorithm for an optimal harmonious coloring of such a forest.Comment: 8 pages, 1 figur
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