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 Karo\'nski, uczak 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 = (), 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.