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) ≤ β∗(G), the fractional vertex cover number. Finally, the application of bc∗(G) and bp∗(G) to two different weak products is discussed.