Edge-coloring Vertex-weightings of Graphs

Let $G=(V(G),E(G))$ be a simple, finite and undirected graph of order $n$. A $k$-vertex weightings of a graph $G$ is a mapping $w: V(G) to {1, ldots, k}$. A $k$-vertex weighting induces an edge labeling $f_w: E(G) to N$ such that $f_w(uv)=w(u)+w(v)$. Such a labeling is called an {it edge-coloring k-vertex weightings} if $f_{w}(e)not= f_{w}(echr(chr(chr('39')39chr('39'))39chr(chr('39')39chr('39'))))$ for any two adjacent edges $e$ and $echr(chr(chr('39')39chr('39'))39chr(chr('39')39chr('39')))$. Denote by $muchr(chr(chr('39')39chr('39'))39chr(chr('39')39chr('39')))(G)$ the minimum $k$ for $G$ to admit an edge-coloring $k$-vertex weightings. In this paper, we determine $muchr(chr(chr('39')39chr('39'))39chr(chr('39')39chr('39')))(G)$ for some classes of graphs.