Fractional biclique covers and partitions of graphs

A biclique is a complete bipartite subgraph of a graph. This paper investigates the fractional biclique cover number, bc*(G), and the fractional biclique partition number,bp*(G), of a graph G. It is observed that bc*(G)and bp*(G) provide lower bounds on the biclique cover and partition numbers respectively, and conditions for equality are given. It is also shown that bc*(G) is a better lower bound on the Boolean rank of a binary matrix than the maximum number of isolated ones of the matrix. In addition, it is noted that bc*(G) <= bp*(G) <= beta*(G), the fractional vertex cover number. Finally, the application of bc*(G) and bp*(G)to two different weak products is discussed.