Document Type

Article

Publication Date

May 2018

Publication Title

Discussiones Mathematicae Graph Theory

DOI

10.7151/dmgt.2156

Keywords

lucky labeling, additive coloring, reducible configuration, discharging method, Combinatorial Nullstellensatz

Disciplines

Discrete Mathematics and Combinatorics | Mathematics

Abstract

An additive coloring of a graph G is a labeling of the vertices of G from {1,2,...,k} such that two adjacent vertices have distinct sums of labels on their neighbors. The least integer k for which a graph G has an additive coloring is called the additive coloring number of G, denoted \luck(G). Additive coloring is also studied under the names lucky labeling and open distinguishing. In this paper, we improve the current bounds on the additive coloring number for particular classes of graphs by proving results for a list version of additive coloring. We apply the discharging method and the Combinatorial Nullstellensatz to show that every planar graph G with girth at least 5 has \luck(G)≤ 19, and for girth at least 6, 7, and 26, \luck(G) is at most 9, 8, and 3, respectively. In 2009, Czerwi\acute{\mbox{n}}ski, Grytczuk, and \dot{\mbox{Z}}elazny conjectured that \luck(G) ≤ χ(G), where χ(G) is the chromatic number of G. Our result for the class of non-bipartite planar graphs of girth at least 26 is best possible and affirms the conjecture for this class of graphs.

Comments

This article is in press for the journal Discussiones Mathematicae Graph Theory and can also be found online at this link.

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