On traceable iterated line graph and hamiltonian path index

TL;DR

This paper characterizes graphs whose iterated line graphs contain Hamiltonian paths and provides bounds on the Hamiltonian path index, extending previous work on Hamiltonian cycles in iterated line graphs.

Contribution

It introduces a new characterization for graphs with Hamiltonian paths in their iterated line graphs, expanding understanding beyond Hamiltonian cycles.

Findings

Characterization of graphs with Hamiltonian paths in iterated line graphs

Upper bounds on the Hamiltonian path index of a graph

Extension of previous Hamiltonian cycle results to Hamiltonian paths

Abstract

Xiong and Liu [L. Xiong and Z. Liu, Hamiltonian iterated line graphs, Discrete Math. 256 (2002) 407-422] gave a characterization of the graphs $G$ for which the $n$-th iterated line graph $L^n(G)$ is hamiltonian, for $n\ge2$. In this paper, we study the existence of a hamiltonian path in $L^n(G)$, and give a characterization of $G$ for which $L^n(G)$ has a hamiltonian path. As applications, we use this characterization to give several upper bounds on the hamiltonian path index of a graph.