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.