There are no finite partial cubes of girth more than 6 and minimum degree at least 3

TL;DR

This paper proves that partial cubes with girth greater than 6 cannot be regular graphs of degree 3 or more, and characterizes their structure as even cycles or tree-zone graphs.

Contribution

It establishes a new structural property of partial cubes with large girth, showing they are either even cycles or tree-zone graphs, and relates their parameters through a new formula.

Findings

Partial cubes of girth > 6 are not regular unless they are even cycles.

Every partial cube with girth > 6 has vertices of degree less than 3.

Girth > 6 partial cubes are characterized as tree-zone graphs with specific parameter relations.

Abstract

Partial cubes are graphs isometrically embeddable into hypercubes. We analyze how isometric cycles in partial cubes behave and derive that every partial cube of girth more than 6 must have vertices of degree less than 3. As a direct corollary we get that every regular partial cube of girth more than 6 is an even cycle. Along the way we prove that every partial cube $G$ with girth more than 6 is a tree-zone graph and therefore $2n(G)-m(G)-i(G)+ce(G)=2$ holds, where $i(G)$ is the isometric dimension of $G$ and $ce(G)$ its convex excess.