Cycles and paths in Jacobson graphs

TL;DR

This paper characterizes the Hamiltonian, pancyclic, and Eulerian properties of finite Jacobson graphs, determining the lengths of their longest cycles and paths, thus advancing understanding of their structural features.

Contribution

It provides a complete characterization of Hamiltonian, pancyclic, and Eulerian properties in finite Jacobson graphs and determines the maximum lengths of induced cycles and paths.

Findings

Finite Jacobson graphs are Hamiltonian if and only if they are pancyclic.

All Jacobson graphs with a Hamiltonian cycle or path, or Eulerian tour or trail are classified.

The maximum lengths of induced cycles and paths in finite Jacobson graphs are established.

Abstract

All finite Jacobson graphs with a Hamiltonian cycle or path, or Eulerian tour or trail are determined, and it is shown that a finite Jacobson graph is Hamiltonian if and only if it is pancyclic. Also, the length of the longest induced cycles and paths in finite Jacobson graphs are obtained.