A k-dimensional box is the cartesian product R1 × R2 × ⋯ × Rk where each Ri is a closed interval on the real line. The boxicity of a graph G, denoted as b o x (G), is the minimum integer k such that G is the intersection graph of a collection of k-dimensional boxes. A unit cube in k-dimensional space or a k-cube is defined as the cartesian product R1 × R2 × ⋯ × Rk where each Ri is a closed interval on the real line of the form [ai, ai + 1]. The cubicity of G, denoted as c u b (G), is the minimum k such that G is the intersection graph of a collection of k-cubes. In this paper we show that c u b (G) ≤ t + ⌈ log (n - t) ⌉ - 1 and b o x (G) ≤ ⌊ frac(t, 2) ⌋ + 1, where t is the cardinality of a minimum vertex cover of G and n is the number of vertices of G. We also show the tightness of these upper bounds. F.S. Roberts in his pioneering paper on boxicity and cubicity had shown that for a graph G, b o x (G) ≤ ⌊ frac(n, 2) ⌋ and c u b (G) ≤ ⌈ frac(2 n, 3) ⌉, where n is the number of vertices of G, and these bounds are tight. We show that if G is a bipartite graph then b o x (G) ≤ ⌈ frac(n, 4) ⌉ and this bound is tight. We also show that if G is a bipartite graph then c u b (G) ≤ frac(n, 2) + ⌈ log n ⌉ - 1. We point out that there exist graphs of very high boxicity but with very low chromatic number. For example there exist bipartite (i.e., 2 colorable) graphs with boxicity equal to frac(n, 4). Interestingly, if boxicity is very close to frac(n, 2), then chromatic number also has to be very high. In particular, we show that if b o x (G) = frac(n, 2) - s, s ≥ 0, then χ (G) ≥ frac(n, 2 s + 2), where χ (G) is the chromatic number of G. © 2008 Elsevier B.V. All rights reserved.
Sunil Chandran, L., Das, A., & Shah, C. D. (2009). Cubicity, boxicity, and vertex cover. Discrete Mathematics, 309(8), 2488–2496. https://doi.org/10.1016/j.disc.2008.06.003