On antipodal and diametrical partial cubes

Abstract

We prove that any diametrical partial cube of diameter at most 6 is antipodal. Because any antipodal graph is harmonic, this gives a partial answer to a question of Fukuda and Handa [Antipodal graphs and oriented matroids, Discrete Math. 111 (1993) 245-256] whether any diametrical partial cube is harmonic, and improves a previous result of Klavžar and Kovše [On even and harmonic-even partial cubes, Ars Combin. 93 (2009) 77-86].

Generalization of edge general position problem

“…A graph [4,25,26,29] present a selection of recent developments on partial cubes. Edges xy and uv of a graph G are in relation…”

The Edge General Position Problem

“…A graph [3,19,20,23] present a selection of recent developments on partial cubes. Edges xy and uv of a graph G are in relation…”