Bipartite dimensions and bipartite degrees of graphs
Let \(G=(V,E)\) be an undirected graph with finite nonempty vertex set \(V\) and irreflexive edge set \(E\). The irreflexive complement of \(G\) is denoted by \(G^c=(V,E^c)\) with \(E^c=\{\{u,v\}:u, v\in V, u\neq v, \{u,v\}\not\in E\}\). A cover of a graph \(G\) is a family of complete bipartite subgraphs of \(G\) whose edges cover the edges of \(G\). The bipartite dimension \(d(G)\) of \(G\) is the minimum cardinality of a cover, and its bipartite degree \(\eta(G)\) is the minimum over all covers of the maximum number of covering members incident to a vertex. The authors prove that \(d(G)\) equals the Boolean interval dimension of the irreflexive complement of \(G\), identify the 21 minimal forbidden induced subgraphs for \(d\leq 2\), and investigate the forbidden graphs for \(d\leq n\) that have the fewest vertices. They show that for complete graphs \(K_n\), \(d(K_n)=\lceil\log_2 n\rceil\), \(\eta(K_n)=d(K_n)\) for \(n\leq 16\), and \(\eta(K_n)\) is unbounded. They also show that the list of minimal forbidden induced subgraphs for \(\eta\leq 2\) is infinite. Two infinite families in this list along with all members that have fewer than seven vertices are given.
scientific article; zbMATH DE number 955150
scientific article