Hamiltonian paths in iterated line graphs

TL;DR

This paper introduces the Hamiltonian path index for graphs, establishing its existence for all graphs and determining its value for trees, while exploring related properties in graphs with 2-connected blocks.

Contribution

The paper defines the Hamiltonian path index, proves its existence for all graphs, and calculates its exact value for trees, advancing understanding of Hamiltonian paths in iterated line graphs.

Findings

Hamiltonian path index exists for all graphs.

Exact value of index determined for trees.

Discusses properties in graphs with 2-connected blocks.

Abstract

For integer $n$, the $n$-iterated line graph $L^n(G)$ of an undirected graph $G$ is defined to be $L(L^{n-1}(G))$, where $L^1(G)$ is the line graph $L(G)$ of $G$. In this paper we introduce hamiltonian path index. Hamiltonian path index, denoted by $h_p(G)$, is the minimum number $n$ such that $L^n(G)$ contains a hamiltonian path. We show that hamiltonian path index of $G$ exists for any graph $G$ and we set the exact value of hamiltonian path index for trees and discuss the problem about graphs with hamiltonian 2-connected blocks.