Vertex-coloring 2-edge-weighting of graphs (Q607360)

A \textit{\(k\)-edge-weighting \(w\)} of a graph is an assignment of an integer weight \(w(e) \in \{1,\dots,k\}\) to each edge \(e\). An edge-weighting naturally induces a coloring \(c\) of vertices by defining \(c(u) = \sum_{e \ni u} w(e)\). A \(k\)-edge weighting of \(G\) is a \textit{vertex-coloring} if the induced coloring \(c\) is proper, that is, \(c(u) \neq c(v)\) for any edge \(uv \in E(G)\). Karoński, Luczak and Thomason conjectured that every connected graph with at least 3 vertices admits a vertex-coloring 3-edge-weighting. The conjecture is known to be true for cubic graphs. It is also known that every \(k\)-colorable graph admits a vertex-coloring \(k\)-edge-weighting if \(k\) is odd, or if \(k \equiv 0 \pmod{4}\). In this paper the authors show that if \(G\) is \(k\)-colorable, \(k \equiv 2 \pmod{4}\), \(k \geq 6\), \(G\) is 2-connected, and \(\delta(G) \geq k-1\), then \(G\) admits a vertex-coloring \(k\)-edge-weighting. They also obtain several sufficient conditions for graphs to admit a vertex-coloring \(k\)-edge-weighting.