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

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 $G_1,G_2$ which fulfil the 1–2–3 Conjecture with just weights 1,2, i.e. such that there exist weightings ${\omega }_i:E\left(G_i\right)\to \{1,2\}$ so that for every $uv\in E$, if $uv\in E\left(G_i\right)$ then $d_{{\omega }_i}\left(u\right)\ne d_{{\omega }_i}\left(v\right)$, where $d_{{\omega }_i}\left(v\right)$ denotes the sum of weights incident with $v\in V$ in $G_i$ 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 $\delta \left(G\right)\ge 3660$, then $G$ can be decomposed into graphs $G_1,G_2$ for which weightings ${\omega }_i:E\left(G_i\right)\to \{1,2\}$ exist so that for every $uv\in E$, $d_{{\omega }_1}\left(u\right)\ne d_{{\omega }_1}\left(v\right)$ or $d_{{\omega }_2}\left(u\right)\ne d_{{\omega }_2}\left(v\right)$. In fact we prove a stronger result, as one of the weightings is redundant, i.e. uses just weight 1.     

Fractional chromatic numbers of tensor products of three graphs

The tensor product $(G_1,G_2,G_3)$ of graphs $G_1$, $G_2$ and $G_3$ is defined by $V(G_1,G_2,G_3)=V\left(G_1\right)\times V\left(G_2\right)\times V\left(G_3\right)$and $E(G_1,G_2,G_3)=\left\{\left((u_1,u_2,u_3),(v_1,v_2,v_3)\right):\left|\{i:(u_i,v_i)\in E\left(G_i\right)\}\right|\ge 2\right\}.$Let ${\chi }_f\left(G\right)$ be the fractional chromatic number of a graph $G$. In this paper, we prove that if one of the three graphs $G_1$, $G_2$ and $G_3$ is a circular clique, ${\chi }_f(G_1,G_2,G_3)=min\{{\chi }_f\left(G_1\right){\chi }_f\left(G_2\right),{\chi }_f\left(G_1\right){\chi }_f\left(G_3\right),{\chi }_f\left(G_2\right){\chi }_f\left(G_3\right)\}.$