Bipartite dimensions and bipartite degrees of graphs

Abstract

A cover (bipartite) of a graph G is a family of complete bipartite subgraphs of G whose edges cover G's edges. G's bipartite dimension d(G) is the minimum cardinality of a cover, and its bipartite degree η(G) is the minimum over all covers of the maximum number of covering members incident to a vertex. We prove that d( G) equals the Boolean interval dimension of the irreflexive complement of G, identify the 21 minimal forbidden induced subgraphs for d ⩽ 2, and investigate the forbidden graphs for d ⩽ n that have the fewest vertices. We note that for complete graphs, d( K n ) = [log 2 n], η( K n ) = d( K n ) for n ⩽ 16, and η( K n ) is unbounded. The list of minimal forbidden induced subgraphs for η ⩽ 2 is infinite. We identify two infinite families in this list along with all members that have fewer than seven vertices.