On Even and Harmonic-Even Partial Cubes

Abstract

Fukuda and Handa $[7]$ asked whether every even partial cube $G$ is harmonic-even. It is shown that the answer is positive if the isometric dimension of $G$ equals its diameter which is in turn true for partial cubes with isometric dimension at most $6$. Under an additional technical condition it is proved that an even partial cube $G$ is harmonic-even or has two adjacent vertices whose diametrical vertices are at distance at least $4$. Some related open problems are posed.