Is there a universal constant K, say K = 3, such that one may dispose of all pairs of adjacent vertices with equal degrees from any given connected graph of order at least three by blowing its selected edges into at most K parallel edges? This question was first posed in 2004 by Karoński, Łuczak 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 . We show that this conjecture holds if the minimum degree δ of a graph is large enough compared to its maximum degree Δ, i.e., when δ = Ω(log Δ).
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Przybyło, J. The 1-2-3 Conjecture Holds for Graphs with Large Enough Minimum Degree. Combinatorica 42 (Suppl 2), 1487–1512 (2022). https://doi.org/10.1007/s00493-021-4822-0
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