Bipartite dimensions and bipartite degrees of graphs
type: Item
instance of: scholarly article
title: Bipartite dimensions and bipartite degrees of graphs
zbMATH Open document ID: 0861.05035
DOI: 10.1016/0012-365X(95)00154-O
author: Peter L. Hammer
author: Peter Fishburn
published in: Discrete Mathematics
publication date: 1997-05-04T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 05C35
Mathematics Subject Classification ID: 05C70
zbMATH DE Number: 955150
zbMATH Keywords: bipartite dimension
zbMATH Keywords: cover
zbMATH Keywords: bipartite degree
zbMATH Keywords: Boolean interval dimension
zbMATH Keywords: forbidden graphs
MaRDI profile type: Publication
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full work available at URL: 0012-365x(95)00154-o
OpenAlex ID: W1994836458
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description: scientific article; zbMATH DE number 955150