On density of subgraphs of halved cubes

Abstract

Let S be a family of subsets of a set X of cardinality m and VC-dim(S) be the Vapnik–Chervonenkis dimension of S. Haussler et al. (1994) proved that if G1(S)=(V,E) is the subgraph of the hypercube Qm induced by S (called the 1-inclusion graph of S), then |E||V|≤VC-dim(S). Haussler (1995) presented an elegant proof of this inequality using the shifting operation.In this note, we adapt the shifting technique to prove that if S is an arbitrary set family and G1,2(S)=(V,E) is the 1,2-inclusion graph of S (i.e., the subgraph of the square Qm2 of the hypercube Qm induced by S), then |E||V|≤d2, where d≔cVC-dim∗(S) is the clique-VC-dimension of S (which we introduce in this paper). The 1,2-inclusion graphs are exactly the subgraphs of halved cubes and comprise subgraphs of Johnson graphs as a subclass.