On Biclique Cover and Partition of the Kneser Graph

On Biclique Cover and Partition of the Kneser Graph

Let bcd(G) (resp. bpd(G)) denote the minimum number of bicliques of G such that every edge of G belongs to at least (resp. exactly) d of these bicliques. Hajiabolhassan and Moazami (2012a) showed existence of a secure frame proof code results from the existence of biclique cover of Kneser graph (KG(t, r)) and vice versa. Also, they (Hajiabolhassan and Moazami, 2012a) showed that using d-biclique cover of Kneser graphs, we can obtain appropriate lower and upper bound for the minimum number of points in cover free family. As was shown by Orlin (1977), determining the exact value biclique covering number is NP-hard. Hence, it is a challenging and interesting problem to determine the exact value of bcd(KG(t, r)) (bpd(KG(t, r))). In this paper, we determine the exact value of bcd(KG(t, r)) (bpd(KG(t, r))) for every r, t, where 2r ≤ t and some d.

Let bcd(G) (resp. bpd(G)) denote the minimum number of bicliques of G such that every edge of G belongs to at least (resp. exactly) d of these bicliques. Hajiabolhassan and Moazami (2012a) showed existence of a secure frame proof code results from the existence of biclique cover of Kneser graph (KG(t, r)) and vice versa. Also, they (Hajiabolhassan and Moazami, 2012a) showed that using d-biclique cover of Kneser graphs, we can obtain appropriate lower and upper bound for the minimum number of points in cover free family. As was shown by Orlin (1977), determining the exact value biclique covering number is NP-hard. Hence, it is a challenging and interesting problem to determine the exact value of bcd(KG(t, r)) (bpd(KG(t, r))). In this paper, we determine the exact value of bcd(KG(t, r)) (bpd(KG(t, r))) for every r, t, where 2r ≤ t and some d.