Narrow Results

The lucky edge coloring of graph $G$ is a proper edge coloring which is induced by a vertex coloring such that each edge is labeled by the sum of its vertices. The least integer $k$ for which $G$ has a lucky edge coloring in the set $\{1,2,\dots ,k\}$ is called lucky number, denoted by $\eta (G)$. The lucky numbers were already calculated for a large number of graphs, but not yet for trees. In this paper, we provide the characterization of lucky edge coloring and calculate the lucky number for graphs which can be regarded as complete $m$-ary trees.