The edge-distinguishing chromatic number of spider graphs with three legs or bounded leg lengths

Abstract

The edge-distinguishing chromatic number $\lambda (G)$ of a simple graph $G$ is the minimum number of colors $k$ assigned to the vertices in $V(G)$ such that each edge $\{u_i,u_j\}$ corresponds to a different set $\{c(u_i),c(u_j)\}$. Al-Wahabi et al.\ derived an exact formula for the edge-distinguishing chromatic number of a path and of a cycle. We derive an exact formula for the edge-distinguishing chromatic number of a spider graph with three legs and of a spider graph with $\mathrm{\Delta }$ legs whose lengths are between 2 and $\frac{\mathrm{\Delta }+3}{2}$.