The locating-chromatic number for Halin graphs

Abstract

Let $G$ be a connected graph‎. ‎Let $f$ be a proper $k$-coloring of $G$ and $\Pi=\{R_1,R_2,\ldots‎, ‎R_k\}$ be an ordered partition of $V(G)$ into color classes‎. ‎For any vertex $v$ of $G,$ define the {\em color code} $c_\Pi(v)$ of $v$ with respect to $\Pi$ to be a $k$-tuple $(d(v,R_1),d(v,R_2),\ldots,d(v,R_k)),$ where $d(v,R_i)= \text{min}\{d(v,x)|x\in R_i\}.$ If distinct vertices have distinct color codes‎, ‎then we call $f$ a {\em locating coloring} of $G.$ The {\em locating-chromatic number} of $G$ is the minimum number $k$ such that $G$ admits a locating coloring with $k$ colors.‎ ‎In this paper‎, ‎we determine a lower bound of the locating-chromatic number of Halin graphs.‎ ‎We also give the locating-chromatic number of a Halin graph of a double star.

Keywords

• locating-chromatic number‎
• ‎Halin graph‎
• ‎double star