A characterization of regular partial cubes whose all convex cycles have the same lengths

TL;DR

This paper characterizes regular partial cubes with all convex cycles of the same length, showing they are hypercubes, doubled odd graphs, or even cycles, thus generalizing known results about median graphs.

Contribution

It provides a complete characterization of regular partial cubes with uniform convex cycle lengths, linking them to well-known graph classes and generalizing previous theorems.

Findings

Regular partial cubes with all convex cycles of length 4 are almost-median graphs.

Such graphs are hypercubes, doubled odd graphs, or even cycles of length 2n.

Regular almost-median graphs are exactly hypercubes.

Abstract

Partial cubes are graphs that can be isometrically embedded into hypercubes. Convex cycles play an important role in the study of partial cubes. In this paper, we prove that a regular partial cube is a hypercube (resp., a Doubled Odd graph, an even cycle of length $2n$ where $n\geqslant 4$) if and only if all its convex cycles are 4-cycles (resp., 6-cycles, $2n$-cycles). In particular, the partial cubes whose all convex cycles are 4-cycles are equivalent to almost-median graphs. Therefore, we conclude that regular almost-median graphs are exactly hypercubes, which generalizes the result by Mulder [J. Graph Theory, 4 (1980) 107--110] -- regular median graphs are hypercubes.