On Some Characterizations of Antipodal Partial Cubes

Abstract

We prove that any harmonic partial cube is antipodal, which was conjectured by Fukuda and K. Handa, Antipodal graphs and oriented matroids, Discrete Math. 111 (1993) 245–256. Then we prove that a partial cube G is antipodal if and only if the subgraphs induced by Wab and Wba are isomorphic for every edge ab of G. This gives a positive answer to a question of Klavžar and Kovše, On even and harmonic-even partial cubes, Ars Combin. 93 (2009) 77–86. Finally we prove that the distance-balanced partial cube that are antipodal are those whose pre-hull number is at most 1.

Keywords

• diametrical graph
• harmonic graph
• antipodal graph
• distance-balanced graph
• partial cube
• pre-hull number
• 05c12
• 05c75