The 1-2-3 Conjecture, posed in 2004 by Karonski, Luczak, and Thomason, is as follows:" If G
is a graph with no connected component having exactly 2 vertices, then the edges of G may …

A solution to the 1-2-3 conjecture - ScienceDirect Skip to main contentSkip to article Elsevier
logo Journals & Books Search RegisterSign in View PDF Download full issue Search …

A weighting of the edges of a graph is called irregular if the weighted degrees of the vertices
are all different. In this note we show that such a weighting is possible from the weight set {1 …

The history, formulas, and most famous puzzles of graph theory Graph theory goes back
several centuries and revolves around the study of graphs—mathematical structures …

Consider a graph G=(V,E) of minimum degree δ and order n. Its irregularity strength is the
smallest integer k for which one can find a weighting w:E→{1,2,...,k\} such that …

This paper is dedicated to studying the following question: Is it always possible to injectively
assign the weights 1,...,| E (G)| to the edges of any given graph G (with no component …

Graph labelings have existed for well over a century. In fact, a graph G is often defined as a
finite nonempty set V of objects called vertices and a set E of 2-element subsets of V called …

Let k be an odd natural number≥ 5, and let G be a (6 k− 7)-edge-connected graph of
bipartite index at least k− 1. Then, for each mapping f: V (G)→ N, G has a subgraph H such …

We consider a proper coloring c of edges and vertices in a simple graph and the sum f (v) of
colors of all the edges incident to v and the color of a vertex v. We say that a coloring c …

A locally irregular graph is a graph whose adjacent vertices have distinct degrees. We say
that a graph G can be decomposed into k locally irregular subgraphs if its edge set may be …