Netlike partial cubes I. General properties

A set A of vertices of a graph G is C-convex if the vertex set of any cycle of the subgraph of G induced by the union of the intervals between each pair of elements of A is contained in A. A partial cube (isometric subgraph of a hypercube) is a netlike partial cube if, for each edge ab, the sets U_ab and U_ba are C-convex (U_ab being the set of all vertices closer to a than to b and adjacent to some vertices closer to b than to a, and vice versa for U_ba). Particular netlike partial cubes are median graphs, even cycles, benzenoid graphs and cellular bipartite graphs. In this paper we give different characterizations and properties of netlike partial cubes. In particular, as median graphs and cellular bipartite graphs, these graphs have a pre-hull number which is at most one, and moreover the convex hull of any isometric cycle of a netlike partial cube is, as in the case of bipartite cellular graphs, this cycle itself or, as in the case of median graphs, a hypercube. We also characterize the gated subgraphs of a netlike partial cube, and we show that the gated amalgam of two netlike partial cubes is a netlike partial cube.     

Netlike partial cubes III. The median cycle property

A graph G has the Median Cycle Property (MCP) if every triple (u₀,u₁,u₂) of vertices of G admits a unique median or a unique median cycle, that is a gated cycle C of G such that for all i,j,k∈{0,1,2}, if x_i is the gate of u_i in C, then: {x_i,x_j}⊆I_G(u_i,u_j) if i≠j, and d_G(x_i,x_j)<d_G(x_i,x_k)+d_G(x_k,x_j). We prove that a netlike partial cube has the MCP if and only if it contains no triple of convex cycles pairwise having an edge in common and intersecting in a single vertex. Moreover a finite netlike partial cube G has the MCP if and only if G can be obtained from a set of even cycles and hypercubes by successive gated amalgamations, and equivalently, if and only if G can be obtained from K₁ by a sequence of special expansions. We also show that the geodesic interval space of a netlike partial cube having the MCP is a Pash-Peano space (i.e. a closed join space).     

Netlike partial cubes II. Retracts and netlike subgraphs

First we show that the class of netlike partial cubes is closed under retracts. Then we prove, for a subgraph G of a netlike partial cube H, the equivalence of the assertions: G is a netlike subgraph of H; G is a hom-retract of H; G is a retract of H. Finally we show that a non-trivial netlike partial cube G, which is a retract of some bipartite graph H, is also a hom-retract of H if and only if G contains at most one convex cycle of length greater than 4.     

Partial cubes: structures, characterizations, and constructions

Partial cubes are isometric subgraphs of hypercubes. Structures on a graph defined by means of semicubes, and Djokovi?’s and Winkler’s relations play an important role in the theory of partial cubes. These structures are employed in the paper to characterize bipartite graphs and partial cubes of arbitrary dimension. New characterizations are established and new proofs of some known results are given.The operations of Cartesian product and pasting, and expansion and contraction processes are utilized in the paper to construct new partial cubes from old ones. In particular, the isometric and lattice dimensions of finite partial cubes obtained by means of these operations are calculated.     

New partial geometric difference sets and partial geometric difference families

Using Galois rings and Galois fields, we construct several infinite classes of partial geometric difference sets, and partial geometric difference families, with new parameters. Furthermore, these partial geometric difference sets (and partial geometric difference families) correspond to new infinite families of directed strongly regular graphs. We also discuss some of the links between partially balanced designs, 2-adesigns (which were recently coined by Cunsheng Ding in “Codes from Difference Sets” (2015)), and partial geometric designs, and make an investigation into when a 2-adesign is a partial geometric design.