Vertex-coloring Edge-weightings of Graphs
Gerard J. Chang, Changhong Lü, Jiaojiao Wu, Qinglin Yu · Taiwanese Journal of Mathematics · 2011
A $k$-edge-weighting of a graph $G$ is a mapping $w: E(G) \to \{1,2,\ldots, k\}$. An edge-weighting $w$ induces a vertex coloring $f_w: V(G) \to \mathbb{N}$ defined by $f_w(v) = \sum_{v \in e} w(e)$. An edge-weighting $w$ is vertex-coloring if $f_w(u) e f_w(v)$ for any edge $uv$. The current paper studies the parameter $\mu(G)$, which is the minimum $k$ for which $G$ has a vertex-coloring $k$-edge-weighting. Exact values of $\mu(G)$ are determined for several classes of graphs, including trees and $r$-regular bipartite graph with $r \ge 3$.