# Hamiltonian chromatic number of block graphs

## TL;DR

This paper investigates the hamiltonian chromatic number of block graphs, providing a necessary and sufficient condition for a lower bound, and presents an algorithm for optimal coloring in specific classes of these graphs.

## Contribution

It establishes a precise condition for the lower bound of the hamiltonian chromatic number of block graphs and introduces an algorithm for optimal coloring in certain block graph classes.

## Key findings

- Derived a necessary and sufficient condition for the lower bound of hc(G) in block graphs.
- Developed an algorithm for optimal hamiltonian coloring of SDB(p/2) block graphs.
- Characterized level-wise regular block graphs and extended star of blocks achieving the lower bound.

## Abstract

Let $G$ be a simple connected graph of order $n$. A hamiltonian coloring $c$ of a graph $G$ is an assignment of colors (non-negative integers) to the vertices of $G$ such that $D(u, v)$ + $|c(u) - c(v)|$ $\geq$ $n - 1$ for every two distinct vertices $u$ and $v$ of $G$, where $D(u, v)$ denotes the detour distance between $u$ and $v$ in $G$ which is the length of the longest path between $u$ and $v$. The value \emph{hc(c)} of a hamiltonian coloring $c$ is the maximum color assigned to a vertex of $G$. The hamiltonian chromatic number, denoted by $hc(G)$, is min\{$hc(c)$\} taken over all hamiltonian coloring $c$ of $G$. In this paper, we give a necessary and sufficient condition to achieve a lower bound for the hamiltonian chromatic number of block graphs given in [Theorem 1,On Hamiltonian Colorings of Block graphs, In: Kaykobad, M., Petrechi, R., (eds.) WALCOM: Algorithms and Computation, LNCS: 9627, 28-39, 2016]. We present an algorithm for optimal hamiltonian coloring of a special class of block graphs, namely $SDB(p/2)$ block graphs. We characterize level-wise regular block graphs and extended star of blocks achieving this lower bound.