Bipartite Graphs with Balanced \((a, b)\)-Partitions

In this paper, we will be concerned with graphs \(G\) satisfying: \(G\) is isometrically embeddable in a hypercube; \(|C(a,b)| = |C(b,a)|\) for every edge \([a,b]\) of \(G\). where \(C(a,b)\) is the set of vertices nearer to \(a\) than to \(b\). Some properties of such graphs are shown; in particular, it is shown that all such graphs \(G\) are \(3\)-connected if \(G\) has at least two edges and \(G\) is not a cycle.