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].

Keywords

• diametrical graph
• harmonic graph
• antipodal graph
• partial cube
• diameter
• isometric dimension
• 05c12
• 05c75