Hamiltonian chromatic number of trees

Let $G$ be a simple finite connected graph of order $n$. The detour distance between two distinct vertices $u$ and $v$ denoted by $D(u,v)$ is the length of a longest $uv$-path in $G$. A hamiltonian coloring $h$ of a graph $G$ of order $n$ is a mapping $h:V(G)\to \{0,1,2,\mathrm{.}\mathrm{.}\mathrm{.}\}$ such that $D(u,v) + |h(u)-h(v)| \geq n-1$,foreverytwodistinctvertices$u$and$v$of$G$. The span of $h$, denoted by $span(h)$, is $\max \{\mathrm{\mid }h(u)-h(v)\mathrm{\mid }:u,v\in V(G)\}$. The hamiltonian chromatic number of $G$ is defined as $hc(G) := \min{span(h)}$withminimumtakenoverallhamiltoniancoloring$h$of$G$. In this paper, we give an improved lower bound for the hamiltonian chromatic number of trees and give a necessary and sufficient condition to achieve the improved lower bound. Using this result, we determine the hamiltonian chromatic number of two families of trees.