Classification of vertex-transitive cubic partial cubes

TL;DR

This paper classifies all cubic, vertex-transitive partial cubes, identifying a specific set of graphs that encompass all such structures, thereby advancing the understanding of their combinatorial and geometric properties.

Contribution

It provides a complete classification of cubic, vertex-transitive partial cubes, extending previous results and including all cubic, distance-regular partial cubes.

Findings

Identifies specific graphs isomorphic to all cubic, vertex-transitive partial cubes.

Extends classification to include all cubic, distance-regular partial cubes.

Generalizes previous results on cubic mirror graphs.

Abstract

Partial cubes are graphs isometrically embeddable into hypercubes. In this paper it is proved that every cubic, vertex-transitive partial cube is isomorphic to one of the following graphs: $K_2 \, \square \, C_{2n}$, for some $n\geq 2$, the generalized Petersen graph $G(10,3)$, the cubic permutahedron, the truncated cuboctahedron, or the truncated icosidodecahedron. This classification is a generalization of results of Bre\v{s}ar et al.~from 2004 on cubic mirror graphs, it includes all cubic, distance-regular partial cubes (Weichsel, 1992), and presents a contribution to the classification of all cubic partial cubes.