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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, Littlestone, and Warmuth (Inf. Comput., 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 (J. Combin. Th. A, 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 Q 2 m of the hypercube Qm induced by S), then |E| |V | ≤ d 2 , 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.
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Submitted on : Wednesday, May 20, 2020 - 12:23:03 PM
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Victor Chepoi, Arnaud Labourel, Sébastien Ratel. On density of subgraphs of halved cubes. European Journal of Combinatorics, Elsevier, 2019, 80, pp.57-70. ⟨10.1016/j.ejc.2018.02.039⟩. ⟨hal-02268756⟩

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