Additive List Coloring of Planar Graphs with Given Girth

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 χ_Σ (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 χ_Σ (G) ≤ 19, and for girth at least 6, 7, and 26, χ_Σ (G) is at most 9, 8, and 3, respectively. In 2009, Czerwiński, Grytczuk, and Żelazny conjectured that χ_Σ (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.