Hamiltonicity of 3-arc graphs

TL;DR

This paper proves that connected 3-arc graphs and their iterative versions are Hamiltonian, confirming a significant case of the conjecture that most vertex-transitive graphs are Hamiltonian.

Contribution

It establishes Hamiltonicity of connected 3-arc graphs and their iterative forms, advancing understanding of Hamiltonian properties in complex graph families.

Findings

Connected 3-arc graphs are Hamiltonian.

Iterative 3-arc graphs of connected graphs with minimum degree ≥ 3 are Hamiltonian.

Vertex-transitive graphs isomorphic to 3-arc graphs of arc-transitive graphs are Hamiltonian.

Abstract

An arc of a graph is an oriented edge and a 3-arc is a 4-tuple $(v,u,x,y)$ of vertices such that both $(v,u,x)$ and $(u,x,y)$ are paths of length two. The 3-arc graph of a graph $G$ is defined to have vertices the arcs of $G$ such that two arcs $uv, xy$ are adjacent if and only if $(v,u,x,y)$ is a 3-arc of $G$. In this paper we prove that any connected 3-arc graph is Hamiltonian, and all iterative 3-arc graphs of any connected graph of minimum degree at least three are Hamiltonian. As a consequence we obtain that if a vertex-transitive graph is isomorphic to the 3-arc graph of a connected arc-transitive graph of degree at least three, then it is Hamiltonian. This confirms the well known conjecture, that all vertex-transitive graphs with finitely many exceptions are Hamiltonian, for a large family of vertex-transitive graphs. We also prove that if a graph with at least four vertices is Hamilton-connected, then so are its iterative 3-arc graphs.