A note on the weak (2,2)-Conjecture

Abstract

Let G=(V,E) be any graph without isolated edges. The well known 1–2–3 Conjecture asserts that the edges of G can be weighted with 1,2,3 so that adjacent vertices have distinct weighted degrees, i.e. the sums of their incident weights. It was independently conjectured that if G additionally has no isolated triangles, then it can be edge decomposed into two subgraphs G1,G2 which fulfil the 1–2–3 Conjecture with just weights 1,2, i.e. such that there exist weightings ωi:E(Gi)→{1,2} so that for every uv∈E, if uv∈E(Gi) then dωi(u)≠dωi(v), where dωi(v) denotes the sum of weights incident with v∈V in Gi for i=1,2. We apply the probabilistic method to prove that the known weakening of this so-called Standard (2,2)-Conjecture holds for graphs with minimum degree large enough. Namely, we prove that if δ(G)≥3660, then G can be decomposed into graphs G1,G2 for which weightings ωi:E(Gi)→{1,2} exist so that for every uv∈E, dω1(u)≠dω1(v) or dω2(u)≠dω2(v). In fact we prove a stronger result, as one of the weightings is redundant, i.e. uses just weight 1.