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 [30]. 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).

是否存在一个通用常数K，例如K = 3，这样可以通过将其选定的边吹入至多K平行边？这个问题由 Karoński、Łuczak 和 Thomason 于 2004 年首次提出，他们同样询问是否可以为每个此类图的边分配权重 1、2、3，以便相邻顶点获得不同的加权度——它们的事件权重之和。这个基本问题现在通常被称为 1-2-3 猜想，并已在多篇论文中得到解决。到目前为止，已知权重 1、2、3、4,5 就足够了 [30]。我们表明，如果图的最小度 δ 与其最大度Δ相比足够大，即当δ = Ω (log Δ ) 时，该猜想成立。