Cubicity, boxicity, and vertex cover

Abstract

A k -dimensional box is the cartesian product R 1 × R 2 × ⋯ × R k where each R i 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 R 1 × R 2 × ⋯ × R k where each R i is a closed interval on the real line of the form [ a i , a i + 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 ) ≤ ⌊ 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 ) ≤ ⌊ n 2 ⌋ and c u b ( G ) ≤ ⌈ 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 ) ≤ ⌈ n 4 ⌉ and this bound is tight. We also show that if G is a bipartite graph then c u b ( G ) ≤ 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 n 4 . Interestingly, if boxicity is very close to n 2 , then chromatic number also has to be very high. In particular, we show that if b o x ( G ) = n 2 − s , s ≥ 0 , then χ ( G ) ≥ n 2 s + 2 , where χ ( G ) is the chromatic number of G .