Lucky Choice Number of Planar Graphs with Given Girth

Suppose the vertices of a graph G are labeled with real numbers. For each vertex v ∈ G, let S(v) denote the sum of the labels of all vertices adjacent to v. A labeling is called lucky if S(u) 6= S(v) for every pair u and v of adjacent vertices in G. The least integer k for which a graph G has a lucky labeling from {1, 2, . . . , k} is called the lucky number of the graph, denoted η(G). In 2009, Czerwiński, Grytczuk, and Żelazny [6] conjectured that η(G) ≤ χ(G), where χ(G) is the chromatic number of G. In this paper, we improve the current bounds for particular classes of graphs with a strengthening of the results through a list lucky labeling. We apply the discharging method and the Combinatorial Nullstellensatz to show that for a planar graph G of girth at least 26, η(G) ≤ 3. This proves the conjecture for non-bipartite planar graphs of girth at least 26. We also show that for girth at least 7, 6, and 5, η(G) is at most 8, 9, and 19, respectively.