The 1-2-3 Conjecture holds for graphs with large enough minimum degree

Abstract

A simple graph more often than not contains adjacent vertices with equal degrees. This in particular holds for all pairs of neighbours in regular graphs, while a lot such pairs can be expected e.g. in many random models. Is there a universal constant K, say K = 3, such that one may always dispose of such pairs from any given connected graph with at least three vertices by blowing its selected edges into at most K parallel edges? This question was first posed in 2004 by Karoński, Luczak and Thomason, who equivalently asked if one may assign weights 1, 2, 3 to the edges of every such graph so that adjacent vertices receive distinct weighted degrees -the sums of their incident weights. This basic problem is commonly referred to as the 1-2-3 Conjecture nowadays, and has been addressed in multiple papers. Thus far it is known that weights 1, 2, 3, 4, 5 are sufficient [J. Combin. Theory Ser. B 100 (2010) 347-349]. We show that this conjecture holds if only the minimum degree δ of a graph is large enough, i.e. when δ = Ω(log ∆), where ∆ denotes the maximum degree of the graph. The principle idea behind our probabilistic proof relies on associating random variables with a special and carefully designed distribution to most of the vertices of a given graph, and then choosing weights for major part of the edges depending on the values of these variables in a deterministic or random manner.

Vertex-coloring graphs with 4-edge-weightings

“…For d-regular graphs, an upper bound of k = 4 holds in general [9,3] and the conjecture (i.e., an upper bound of k = 3) is confirmed for d > 10 8 [9]. Recently, Przyby lo verified the conjecture for graphs where the minimum degree is sufficiently large compared to the maximum degree [8]. Furthermore, the conjecture was confirmed for 3-colorable graphs [7] and for dense graphs [12].…”