Neighbor Distinguishing Edge Colorings via the Combinatorial Nullstellensatz

Abstract

Consider a simple graph G = (V, E) and its proper edge coloring c with the elements of the set {1, 2, . . . , k} (or any other k-element set of real numbers). We say that c is neighbor sumWe show that such a coloring exists for any graph G containing no isolated edges if k ≥ 2Δ(G) + col(G) − 1. The proof of this fact is based on iterative applications of the Combinatorial Nullstellensatz. As a consequence, the same number of colors is also sufficient in the well-known corresponding problem, where instead of the sums, we wish to distinguish the sets of colors met by adjacent vertices. In fact we consider list versions of both concepts and prove our assertion in this more general setting.

Asymptotically optimal bound on the adjacent vertex distinguishing edge choice number

“…We denote it by . This was already investigated under different notations for example in (for a few simple graph classes) and in , where it was proved that ≤ 2Δ( G ) + col( G ) − 1 and ≤ Δ( G ) + 3col( G ) − 4 (where col( G ) ≤ Δ( G ) + 1 denotes the coloring number of G , that is, the least integer k such that the vertex set of G can be linearly ordered so that each vertex is preceded by fewer than k its neighbors) for every graph G without isolated edges. The latter of these results implies an upper bound of the form ≤ Δ( G ) + K with a constant K for many classes of graph, for example for planar graphs.…”

“…Analogously as in the case of the classical choosability of graphs, introduced for vertex colorings by Vizing [24] and independently by Erdős, Rubin and Taylor [8], we define the adjacent vertex distinguishing edge choice number of a graph G (without isolated edges) as the least k so that for every set of lists of size k associated to the edges of G we are able to choose colors from the respective lists to obtain an adjacent vertex distinguishing edge coloring of G. We denote it by ch ′ a (G). This was already investigated under different notations for example in [14] (for a few simple graph classes) and in [22,23], where it was proved that ch ′ a (G) ≤ 2Δ(G) + col(G) − 1 and ch ′ a (G) ≤ Δ(G) + 3col(G) − 4 (where col(G) ≤ Δ(G) + 1 denotes the coloring number of G, that is, the least integer k such that the vertex set of G can be linearly ordered so that each vertex is preceded by fewer than k its neighbors) for every graph G without isolated edges. The latter of these results implies an upper bound of the form ch ′ a (G) ≤ Δ(G)+K with a constant K for many classes of graph, for example for planar graphs.…”

Neighbor Distinguishing Edge Colorings Via the Combinatorial Nullstellensatz Revisited

“…We say that the coloring c is neighbor set distinguishing if for every edge uv ∈ E, S c (u) = S c (v). The smallest number of colors in C for which such a coloring exists is called the neighbor set distinguishing index or the adjacent strong chromatic index of a graph, and denoted by χ a (G), see also [3,5,8,9,11,17,20] for different notations. Such a number exists iff G contains no isolated edges.…”

“…Using an algebraic approach we have already shown in [17] that ch (G) ≤ 2 (G) + col(G) − 1, where col(G) denotes the coloring number of G (i.e. the least integer k such that G has a vertex enumeration in which each vertex is preceded by fewer than k of its neighbors, see e.g.…”

Neighbor Sum Distinguishing Index of $$K_4$$ K 4 -Minor Free Graphs

“…Recently, Przybyło [17] proved that nsdi(G) ≤ 2Δ(G)+col(G) − 1, where col(G) is the coloring number of G, which is defined as the least integer k such that G has a vertex enumeration in which each vertex is preceded by fewer than k of its neighbors, hence col(G)−1 ≤ Δ(G), and thus 2Δ(G)+col(G) − 1 ≤ 3Δ(G). Dong et al [9] proved that Conjecture 3 holds for sparse graphs.…”