Secure frameproof codes through biclique covers

Hossein Hajiabolhassan (1, 2) , Farokhlagha Moazami (3)

1 Department of Mathematical Sciences [Tehran]
2 IPM - Institute for Research in Fundamental Science [Terhan]
3 Department of Mathematics [Tehran]

For a binary code Γ of length v, a v-word w produces by a set of codewords {w1,...,wr}⊆Γ if for all i=1,...,v, we have wi∈{w1i,...,wri} . We call a code r-secure frameproof of size t if |Γ|=t and for any v-word that is produced by two sets C1 and C2 of size at most r then the intersection of these sets is nonempty. A d-biclique cover of size v of a graph G is a collection of v-complete bipartite subgraphs of G such that each edge of G belongs to at least d of these complete bipartite subgraphs. In this paper, we show that for t≥2r, an r-secure frameproof code of size t and length v exists if and only if there exists a 1-biclique cover of size v for the Kneser graph KG(t,r) whose vertices are all r-subsets of a t-element set and two r-subsets are adjacent if their intersection is empty. Then we investigate some connection between the minimum size of d-biclique covers of Kneser graphs and cover-free families, where an (r,w;d) cover-free family is a family of subsets of a finite set such that the intersection of any r members of the family contains at least d elements that are not in the union of any other w members. Also, we present an upper bound for 1-biclique covering number of Kneser graphs.

Domaines: Mathématique discrète [cs.DM]

Hossein Hajiabolhassan, Farokhlagha Moazami. Secure frameproof codes through biclique covers. Discrete Mathematics and Theoretical Computer Science, 2012, Vol. 14 no. 2 (2), pp.261--270. ⟨10.46298/dmtcs.585⟩. ⟨hal-00990601⟩