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arXiv · 2012.07375

Hamiltonian chromatic number of trees

Devsi Bantva·Samir Vaidya
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Abstract

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 $h : V(G) \rightarrow \{0,1,2,...\}$ such that $D(u,v) + |h(u)-h(v)| \geq n-1$, for every two distinct vertices $u$ and $v$ of $G$. The span of $h$, denoted by $span(h)$, is $\max\{|h(u)-h(v)| : u, v \in V(G)\}$. The hamiltonian chromatic number of $G$ is defined as $hc(G) := \min\{span(h)\}$ with minimum taken over all hamiltonian coloring $h$ 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.

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Devsi Bantva, Samir Vaidya. 2020-12-14. Hamiltonian chromatic number of trees. https://arxiv.org/abs/2012.07375

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