Sphericity, cubicity, and edge clique covers of graphs

Michael, T. S.; Quint, Thomas

doi:10.1016/j.dam.2006.01.004

Cited by 14 publications

(11 citation statements)

References 18 publications

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“…were shown in [8] by relating the boxicity of a graph with its treewidth. The cube representation of special classes of graphs like hypercubes and complete multipartite graphs were investigated in [19,15,16].…”

Section: Previous Results On Boxicity and Cubicitymentioning

confidence: 99%

Cubicity, degeneracy, and crossing number

Adiga

Chandran

Mathew

2014

European Journal of Combinatorics

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A k-box B = (R 1 , R 2 , . . . , R k ), where each R i is a closed interval on the real line, is defined to be the Cartesian product R 1 × R 2 × · · · × R k . If each R i is a unit length interval, we call B a k-cube. Boxicity of a graph G, denoted as box (G), is the minimum integer k such that G is an intersection graph of k-boxes. Similarly, the cubicity of G, denoted as cub (G), is the minimum integer k such that G is an intersection graph of k-cubes.It was shown in [L. Sunil Chandran, Mathew C. Francis, and Naveen Sivadasan. Representing graphs as the intersection of axis-parallel cubes. MCDES-2008, IISc Centenary Conference, available at CoRR, abs/cs/0607092, 2006.] that, for a graph G with maximum degree ∆, cub(G) ≤ 4(∆ + 1) ln n . In this paper we show that, for a k-degenerate graph G, cub(G) ≤ (k + 2) 2e log n . Since k is at most ∆ and can be much lower, this clearly is a stronger result. We also give an efficient deterministic algorithm that runs in O(n 2 k) time to output a 8k( 2.42 log n + 1) dimensional cube representation for G. The crossing number of a graph G, denoted as CR(G), is the minimum number of crossing pairs of edges, over all drawings of G in the plane. An important consequence of the above result is that if the crossing number of a graph G is t, then box(G) is O(t 1/4 log t 3/4 ) . This bound is tight upto a factor of O((log t)3/4 ). Let (P, ≤) be a partially ordered set and let G P denote its underlying comparability graph. Let dim(P) denote the poset dimension of P. Another interesting consequence of our result is to show that dim(P) ≤ 2(k + 2) 2e log n , where k denotes the degeneracy of G P . Also, we get a deterministic algorithm that runs in O(n 2 k) time to construct a 16k( 2.42 log n + 1) sized realizer for P. As far as we know, though very good upper bounds exist for poset dimension in terms of maximum degree of its underlying comparability graph, no upper bounds in terms of the degeneracy of the underlying comparability graph is seen in the literature. ACM Subject Classification G.2.2 Graph Theory

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Section: Previous Results On Boxicity and Cubicitymentioning

confidence: 99%

Cubicity, degeneracy, and crossing number

Adiga

Chandran

Mathew

2014

European Journal of Combinatorics

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“…The clique cover number has been studied widely in the literature (see e.g. ). In , Chen et al proved that the clique cover number of a quasi‐line graph on

n

vertices is at most

n

.…”

Section: Introductionmentioning

confidence: 99%

Edge clique cover of claw‐free graphs

Javadi

Hajebi

2018

Journal of Graph Theory

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The smallest number of cliques, covering all edges of a graph G, is called the (edge) clique cover number of G and is denoted by cc(G). It is an easy observation that if G is a line graph on n vertices, then cc(G)≤n. G. Chen et al. [Discrete Math. 219 (2000), no. 1–3, 17–26; MR1761707] extended this observation to all quasi‐line graphs and questioned if the same assertion holds for all claw‐free graphs. In this paper, using the celebrated structure theorem of claw‐free graphs due to Chudnovsky and Seymour, we give an affirmative answer to this question for all claw‐free graphs with independence number at least three. In particular, we prove that if G is a connected claw‐free graph on n vertices with three pairwise nonadjacent vertices, then cc(G)≤n and the equality holds if and only if G is either the graph of icosahedron, or the complement of a graph on 10 vertices called “twister” or the pth power of the cycle Cn, for some positive integer p≤⌊(n−1)∕3⌋.

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“…Upper bounds on the boxicity of some special classes of graphs such as chordal graphs, circular arc graphs, AT-free graphs, permutation graphs, co-comparability graphs are given in [4]. The cube representation of special classes of graphs like hypercubes and complete multipartite graphs were investigated in [1,5,10,11,12].…”

Section: Introductionmentioning

confidence: 99%

Cubicity, boxicity, and vertex cover

Chandran

Das

Shah

2009

Discrete Mathematics

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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 box(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 cub(G), is the minimum k such that G is the intersection graph of a collection of k-cubes. In this paper we show that cub(G) ≤ t + ⌈log (n − t)⌉ − 1 and box(G) ≤ t 2 + 1, where t is the cardinality of the 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, box(G) ≤ n 2 , where n is the number of vertices of G, and this bound is tight. We show that if G is a bipartite graph then box(G) ≤ n 4 and this bound is tight. 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 box(G) = n 2 −s, s ≥ 0, then χ(G) ≥ n 2s+2 , where χ(G) is the chromatic number of G.

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Sphericity, cubicity, and edge clique covers of graphs

Cited by 14 publications

References 18 publications

Cubicity, degeneracy, and crossing number

Cubicity, degeneracy, and crossing number

Edge clique cover of claw‐free graphs

Cubicity, boxicity, and vertex cover

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