On the 1-2-3-conjecture

TL;DR

This paper investigates the 1-2-3-conjecture, proving it for certain infinite graph classes and exploring conditions under which graphs admit a vertex-coloring 2-edge weighting.

Contribution

It extends the 1-2-3-conjecture to Cartesian products of graphs and identifies properties for graphs to admit VC2-EW.

Findings

Proved the 1-2-3-conjecture for some infinite classes of graphs.

Explored properties that enable a graph to admit a VC2-EW.

Studied vertex-coloring edge-weightings in Cartesian product graphs.

Abstract

A k-edge-weighting of a graph G is a function w: E(G)->{1,2,...,k}. An edge-weighting naturally induces a vertex coloring c, where for every vertex v in V(G), c(v) is sum of weights of the edges that are adjacent to vertex v. If the induced coloring c is a proper vertex coloring, then w is called a vertex-coloring k-edge weighting (VCk-EW). Karonski et al. (J. Combin. Theory Ser. B 91 (2004) 151-157) conjectured that every graph admits a VC3-EW. This conjecture is known as 1-2-3-conjecture. In this paper, frst, we study the vertex-coloring edge-weighting of the cartesian product of graphs. Among some results, we prove that the 1-2-3-conjecture holds for some infinite classes of graphs. Moreover, we explore some properties of a graph to admit a VC2-EW