en تخصصي Special پژوهشي Research paper <p>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 $f_{w}(e)not= f_{w}(e&#39;)$ for any two adjacent edges $e$ and $e&#39;$. Denote&nbsp;by&nbsp;$mu&#39;(G)$ the minimum $k$ for $G$ to admit an edge-coloring $k$-vertex weightings. In this paper, we determine $mu&#39;(G)$ for some classes of graphs.</p> Edge coloring, Vertex weightings. 1 13 http://ijmsi.ir/browse.php?a_code=A-10-2087-3&slc_lang=en&sid=1 W.-Ch. Shiu wcshiu@hkbu.edu.hk 10031947532846008761 10031947532846008761 No Hong Kong Baptist University G.-Ch. Lau geeclau@yahoo.com 10031947532846008762 10031947532846008762 Yes Universiti Teknologi MARA Malaysia H.-K. Ng ho-kuen.ng@sjsu.edu 10031947532846008763 10031947532846008763 No San Jose State University, USA