A novel characterization of cubic Hamiltonian graphs via the associated quartic graphs

TL;DR

This paper provides a new characterization of cubic Hamiltonian graphs using associated quartic graphs and Eulerian tours, with applications to I-graphs and Cayley multigraphs, advancing understanding of graph Hamiltonicity.

Contribution

It introduces a necessary and sufficient condition for cubic graphs to be Hamiltonian based on associated quartic graphs and extends this to characterize Hamiltonian I-graphs and Cayley multigraphs.

Findings

Characterization of Hamiltonian I-graphs using graph bundles.

Representation of Cayley multigraphs of degree 4 as graph bundles.

Identification of a family of connected cubic (multi)graphs including I-graphs.

Abstract

We give a necessary and sufficient condition for a cubic graph to be Hamiltonian by analyzing Eulerian tours in certain spanning subgraphs of the quartic graph associated with the cubic graph by 1-factor contraction. This correspondence is most useful in the case when it induces a blue and red 2-factorization of the associated quartic graph. We use this condition to characterize the Hamiltonian I-graphs, a further generalization of generalized Petersen graphs. The characterization of Hamiltonian I-graphs follows from the fact that one can choose a 1-factor in any I-graph in such a way that the corresponding associated quartic graph is a graph bundle having a cycle graph as base graph and a fiber and the fundamental factorization of graph bundles playing the role of blue and red factorization. The techniques that we develop allow us to represent Cayley multigraphs of degree 4, that are associated to abelian groups, as graph bundles. Moreover, we can find a family of connected cubic (multi)graphs that contains the family of connected I-graphs as a subfamily.