741 research outputs found
Locally identifying coloring in bounded expansion classes of graphs
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
A harmonious coloring of is a proper vertex coloring of such that
every pair of colors appears on at most one pair of adjacent vertices. The
harmonious chromatic number of , , is the minimum number of colors
needed for a harmonious coloring of . We show that if is a forest of
order with maximum degree , 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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