Additive list coloring of planar graphs with given girth

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 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, andŻelazny conjectured that χ Σ (G) ≤ χ(G), where χ(G) is the chromatic number of G. Our result for

An Alon-Tarsi Style Theorem for Additive Colorings

“…For arbitrary graphs G, it was shown in [2] that η ℓ (G) ≤ ∆ 2 − ∆ + 1, where ∆ ≥ 2 is the maximum degree of G. It was shown later in [5] that η ℓ (G) ≤ k∆ + 1, where k is the degeneracy of G. For planar graphs, the bound given in [5] leads to the upper bound of η ℓ (G) ≤ 5∆ + 1, and another linear bound η ℓ (G) ≤ 2∆ + 25 was proved in [9]. Some constant bounds for planar graphs of given girth can also be found in [7]. Our main result, Theorem 4.2, may have the potential to improve some of these bounds.…”

“…Notably, Czerwiński et al conjectured in [8] that η(G) ≤ χ(G) for all graphs G, where χ(G) is the classical chromatic number. The conjecture remains open, but some progress has been made-for example, it was shown in [6] that if G is planar, η(G) ≤ 468, and more results on the additive chromatic number of planar graphs can be found in [1], [7], and [9]. Progress for other types of graphs can be found in [10], where an up-to-date list of classes of graphs for which the conjecture has been verified is given.…”

On the additive chromatic number of several families of graphs

“…It is known that the conjecture holds for trees [5,7] and, recently, for nonbipartite planar graphs of girth at least 26 [12]. Our contribution in this work is to give the exact value of the additive chromatic number of several families of graphs and expand the number of cases in which the conjecture is satisfied.…”

“…It was first presented by Czerwiński, Grytczuk and Zelazny [5] who proposed a conjecture that for every graph G, η(G) ≤ χ(G), where χ(G) is the chromatic number of G and η(G) is the additive coloring number of G, defined below. The problem as well as the conjecture has recently gained interest from the scientific community [6,7,8,9,10,11,12,13]. However, the additive chromatic number is known for very few families of graphs.…”

“…Constant upper bounds of the additive chromatic number are known for some families of graphs: if G is a tree, η(G) ≤ 2; if G is planar bipartite, η(G) ≤ 3 [5]; if G is planar, η(G) ≤ 468 [11] and if G is planar of girth at least 26, η(G) ≤ 3 [12]. Other upper bounds can be consulted in [5,11,12].…”