Neighbor Distinguishing Edge Colorings Via the Combinatorial Nullstellensatz Revisited

Abstract

Consider a simple graph G = (V, E ) and its proper edge coloring c with the elements of the set {1, . . . , k}. We say that c is neighbor set distinguishing (or adjacent strong) if for every edge uv ∈ E , the set of colors incident with u is distinct from the set of colors incident with v. Let us then consider a stronger requirement and suppose we wish to distinguishing adjacent vertices by sums of their incident colors. In both problems the challenging conjectures presume that such colorings exist for any graph G containing no isolated edges if only k ≥ (G) + 2. We prove that in both problems k = (G) + 3col(G) − 4 is sufficient. The proof is based on the Combinatorial Nullstellensatz, applied in the "sum environment." In fact the identical bound also holds if we use any set of k real numbers instead of {1, . . . , k} as edge colors, and the same is true in list versions of the both concepts. In particular, we therefore obtain that lists of length (G) + 14 ( (G) + 13 in fact) are sufficient for planar graphs.

On the Neighbor Sum Distinguishing Index of Planar Graphs

“…This was preceded by other general upper bounds, involving the coloring number of G , col( G ), 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 ). In particular in and , the bounds and (or even for nontrees), resp., were proved for every graph G containing no isolated edges by means of algebraic tools based on Combinatorial Nullstellensatz by Alon, see . As a consequence of the algebraic approach applied in these two articles, the same upper bounds hold also in a more general list setting of the same concept.…”

Distant sum distinguishing index of graphs with bounded minimum degree

“…Upper bounds of orders conjectured above are also known for r = 2, 3, but with slightly worse multiplicative constants than in Theorem 1.2 above, see [31], while the upper bound of the form χ Σ (G) ≤ (1 + o(1))∆(G) was proved in [29] and [32], see also [6,11,14,28,33,34] for other results concerning the case r = 1. In this paper we combine probabilistic approach with a special constructive algorithm in order to provide the following improvements of the best known upper bounds for all r ≥ 4 from Theorem 1.2, under assumption that the minimum degree of a graph is larger than some poly-logarithmic function of the maximum degree.…”

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.…”