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Grid intersection graphs and boxicity. (English) Zbl 0784.05031

A graph has boxicity \(k\) if \(k\) is the smallest integer such that the graph can be presented as an intersection of parallelepipeds in \(k\)- dimensional space. The bipartite graphs are shown to have boxicity 2. Some inequalities for graphs in higher dimensions are proved.

MSC:

05C35 Extremal problems in graph theory
05C99 Graph theory
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
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References:

[1] Ben-Arroyo Hartman, I.; Newman, Ilan; Ziv, Ran, On grid intersection graphs, Discrete Math., 87, 41-52 (1991) · Zbl 0739.05081
[2] F.S. Roberts, On the boxicity and cubicity of a graph, Recent Progress in Combinatorics (Academic Press, New York).; F.S. Roberts, On the boxicity and cubicity of a graph, Recent Progress in Combinatorics (Academic Press, New York). · Zbl 0193.24301
[3] Cozzens, M. B.; 8\(Roberts, F. S., Computing the boxicity of a graph by covering its complement by cointerval graphs, Discrete. Appl. Math., 6, 217-228 (1983\) · Zbl 0524.05059
[4] Scheinerman, E. R., Irrepresentability by multiple intersection, or why the interval number is unbounded, Discrete Math., 55, 195-211 (1985) · Zbl 0576.05030
[5] Trotter, W. T., A characterization of Roberts’ inequality for boxicity, Discrete Math., 28, 303-313 (1979) · Zbl 0421.05062
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