The 1-2-3 Conjecture and related problems: a survey

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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 be assigned weights from the set 1,2,3 so that, for any adjacent vertices u and v, the sum of weights of edges incident to u differs from the sum of weights of edges incident to v." This survey paper presents the current state of research on the 1-2-3 Conjecture and the many variants that have been proposed in its short but active history.