Upper bound on cubicity in terms of boxicity for graphs of low chromatic number

Discrete Mathematics 339 (2016) 443–446

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Upper bound on cubicity in terms of boxicity for graphs of low chromatic number L. Sunil Chandran a , Rogers Mathew b , Deepak Rajendraprasad c,∗ a

Department of Computer Science and Automation, Indian Institute of Science, Bangalore, 560012, India

b

Department of Computer Science and Engineering, Indian Institute of Technology Kharagpur, 721302, India

c

The Caesarea Rothschild Institute, Department of Computer Science, University of Haifa, 31095, Haifa, Israel

article

info

Article history: Received 23 May 2014 Received in revised form 4 September 2015 Accepted 7 September 2015

Keywords: Boxicity Cubicity Chromatic number

abstract The boxicity (respectively cubicity) of a graph G is the least integer k such that G can be represented as an intersection graph of axis-parallel k-dimensional boxes (respectively kdimensional unit cubes) and is denoted by box(G) (respectively cub(G)). It was shown by Adiga and Chandran (2010) that for any graph G, cub(G) ≤ box(G) ⌈log2 α(G)⌉, where α(G) is the maximum size of an independent set in G. In this note we show that cub(G) ≤ 2 ⌈log2 χ (G)⌉ box(G) + χ (G) ⌈log2 α(G)⌉, where χ (G) is the chromatic number of G. This result can provide a much better upper bound than that of Adiga and Chandran for graph classes with bounded chromatic number. For example, for bipartite graphs we obtain cub(G) ≤ 2(box(G) + ⌈log2 α(G)⌉). Moreover, we show that for every positive integer k, there exist graphs with chromatic number k such that for every ϵ > 0, the value given by our upper bound is at most (1 + ϵ) times their cubicity. Thus, our upper bound is almost tight. © 2015 Published by Elsevier B.V.

1. Introduction A graph G is an intersection graph of sets from a family of sets F , if there exists f : V (G) → F such that uv ∈ E (G) ⇔ f (u) ∩ f (v) ̸= ∅. An interval graph is an intersection graph in which the set assigned to each vertex is a closed interval on the real line. In other words, an interval graph is an intersection graph of closed intervals on the real line. Similarly, a unit interval graph is an intersection graph of closed unit intervals on the real line. An axis-parallel k-dimensional box, abbreviated to k-box, is a Cartesian product of the form R1 × · · · × Rk , where each Ri is an interval of the form [ai , bi ] on the real line. A k-cube is a k-box such that each Ri is an interval of the form [ai , ai + 1]. Given a graph G that is an intersection graph of k-boxes (respectively k-cubes), we call a function f a k-box representation (respectively k-cube representation) of G if f is a function that maps the vertices of G to k-boxes (respectively k-cubes) such that for any two vertices u, v ∈ V (G), it holds that uv ∈ E (G) if and only if f (u) ∩ f (v) ̸= ∅. The boxicity (respectively cubicity) of a graph G, denoted by box(G) (respectively cub(G)), is the minimum non-negative integer k such that G has a k-box representation (respectively k-cube representation). Only complete graphs have boxicity (cubicity) 0. The class of graphs with boxicity at most 1 is the class of interval graphs, and the class of graphs with cubicity at most 1 is the class of unit interval graphs. When H1 and H2 are graphs such that V (H1 ) = V (H2 ) = V (G) and E (G) = E (H1 ) ∩ E (H2 ), we write G = H1 ∩ H2 . The following observation was made by F.S. Roberts [7].

∗

Corresponding author. E-mail addresses: [email protected] (L.S. Chandran), [email protected] (R. Mathew), [email protected] (D. Rajendraprasad).

http://dx.doi.org/10.1016/j.disc.2015.09.007 0012-365X/© 2015 Published by Elsevier B.V.

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L.S. Chandran et al. / Discrete Mathematics 339 (2016) 443–446

Lemma 1 ([7]). The boxicity of a non-complete graph G equals the least k such that there exist interval graphs I1 , . . . , Ik such that G = I1 ∩ · · · ∩ Ik . The cubicity of a non-complete graph G equals the least k such that there exist unit interval graphs U1 , . . . , Uk such that G = U1 ∩ · · · ∩ Uk . In the rest of the paper, we shall use n to denote the number of vertices of the graph being discussed. Logarithms will be to the base 2, unless otherwise specified. Clearly, for any graph G, box(G) ≤ cub(G). This suggests the following question: does there exist a function g such that cub(G) ≤ g (box(G))? It is easy to see that the answer is negative: consider a star on n + 1 vertices. Its cubicity is ⌈log n⌉ [7] whereas its boxicity is 1, since a star is an interval graph. Chandran and Mathew [5] showed that for any graph G, cub(G) ≤ box(G) ⌈log n⌉, where n is the number of vertices. Adiga and Chandran [3] improved this result by showing that n can be replaced by the size α(G) of a maximum independent set in G. Lemma 2 ([3]). For any graph G, cub(G) ≤ box(G) ⌈log α(G)⌉. Remark. We demonstrate next that the bound in the above lemma is tight, i.e., given any two integers b and α , where b ≥ 1 and α ≥ 2, we show that there exists a graph G with box(G) = b, α(G) = α and cub(G) = b⌈log α⌉. It was shown by Roberts in [7] that for p, n1 , . . . , np ≥ 2, the complete p-partite graph with ni vertices in the ith part has boxicity p p and cubicity i=1 ⌈log ni ⌉. Therefore, when b ≥ 2 the complete b-partite graph with α vertices in each part will serve our purpose. If b = 1, then a star with α + 1 vertices has the desired properties, as mentioned above (see [7]). In a loose sense, the two factors in the upper bound on cub(G) given in Lemma 2, namely ⌈log α(G)⌉ and box(G), individually can make the cubicity of a graph high. Clearly box(G) is a lower bound for cub(G) since cubes are specialized boxes. The other term ⌈log α(G)⌉ can make cub(G) high due to a geometric reason, captured in the so-called ‘volume argument’, which we reproduce here (also see [6]): let cub(G) = k and let d be the diameter of G. By considering the extreme intervals when a cube representation in k dimensions is projected in a fixed dimension, the projection in each dimension is contained in an interval of length d + 1. Hence the volume of a representation isat most (d + 1)k . Also the volume is at least  α(G), since it contains that many disjoint cubes. Hence, cub(G) ≥ logd+1 α(G) . Thus M = max(box(G), logd+1 α(G) ) ≤ cub(G). It is natural to ask whether there exists a function g such that cub(G) ≤ g (M ). The answer is no, since we can increase the diameter of a graph without bound without affecting its cubicity. For example, if G is the graph obtained by identifying one endpoint of a path on 2n + 1 vertices with the leaf of a star on n + 1 vertices, it is easy to check that box(G) = 1, α(G) = 2n, the diameter d of G equals 2n + 2 and hence M = max{box(G), ⌈log α(G)/ log(d + 1)⌉} = 1, whereas cub(G) = ⌈log n⌉, which is far higher. (We have noted that the cubicity of a star with n + 1 vertices is ⌈log n⌉ [7], and identifying the endpoint of a path with a leaf of the star does not increase its cubicity.) ¯ = max(box(G), ⌈log α(G)⌉). Lemma 2 tells us that cub(G) ∈ O(M ¯ 2 ), and In this paper we ask a simpler question: let M b the remark after the lemma indicates that we cannot have anything better in general (choosing α = 2 there illustrates ¯ ) for some restricted graph classes? In this paper we show that if we restrict the point). Can we show that cub(G) ∈ O(M ourselves to classes of graphs whose chromatic number is bounded above by a constant, then such a result can indeed be proved. Our main theorem is a general upper bound for cubicity in terms of boxicity, the independence number, and the chromatic number:





Theorem 3. If G is a graph with chromatic number χ (G) and independence number α(G), then cub(G) ≤ 2 ⌈log χ (G)⌉ box(G)+ χ(G) ⌈log α(G)⌉. The proof of Theorem 3 is in Section 2. For graphs of low chromatic number, this result can be in general far better than that of Adiga et al. [3]. The most interesting case is that of bipartite graphs: Corollary 4. For a bipartite graph G, cub(G) ≤ 2(box(G) + ⌈log α(G)⌉). Remark. The reader may wonder whether the chromatic number is an upper bound for the boxicity of a graph, in which case Theorem 3 cannot be an improvement over Lemma 2. In fact, most graphs with fixed chromatic number have larger boxicity. In [2] it is shown that almost all balanced bipartite graphs (on 2n vertices) have boxicity Ω (n). The proof can be modified to show that almost all bipartite graphs with n vertices on one side and m vertices on the other have boxicity Ω (min(n, m)). Using the ideas from [2], it can be proved without much difficulty that, for any fixed k, boxicity is much greater than k for almost all balanced k-partite graphs. It also follows from [2] and [4] that almost all graphs have boxicity much larger than their chromatic number. 1.1. Preliminaries A graph G is a co-bipartite graph if its complement G is a bipartite graph. Thus G is a co-bipartite graph if and only if the vertex set V (G) can be partitioned into two cliques A and B. It is clear that α(G) ≤ 2 when G is co-bipartite. Lemma 2 yields the following lemma.

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Lemma 5. For a co-bipartite graph G, cub(G) = box(G). Lemma 6 ([1]). Let (A, B) be a partition of the vertex set of a graph G. If H is obtained from G by adding edges to turn A and B into cliques, then box(H ) ≤ 2box(G). 2. Proof of Theorem 3 Let k = χ (G). We first obtain, from a proper k-coloring φ of G, ⌈log k⌉ co-bipartite supergraphs of G. Their intersection is not G, because nonadjacent vertices having the same color under φ will be adjacent in each supergraph. Let l = ⌈log k⌉. A standard construction encodes the colors as binary strings of length l to yield l bipartitions of the vertices of G, where v is in Ai or Bi when φ(v) has 0 or 1 in the ith coordinate, respectively. Let Hi be the co-bipartite supergraph of G obtained by adding edges to turn each of Ai and Bi into a clique. Observation 7. For the co-bipartite graph Hi of G defined above, there are 2box(G) unit interval graphs Ui1 , . . . , Ui2b whose intersection is Hi . Proof. Since Hi is co-bipartite, Lemma 5 yields cub(Hi ) = box(Hi ). Lemma 6 implies box(Hi ) ≤ 2box(G). Hence, cub(Hi ) ≤ 2box(G). By definition (or formally what is stated as Lemma 1), the desired unit interval graphs exist.  Let C0 , . . . , Ck−1 be the color classes under the optimal coloring φ , and let ti = ⌈log |Ci |⌉. We define unit interval graphs t Wi1 , . . . , Wi i in the following way: first number the vertices of Ci from 0 to |Ci | − 1. Let ni (u) be the number given to a vertex j

u ∈ Ci by this numbering. For 1 ≤ j ≤ ti , associate with each vertex v ∈ V (G) an interval fi (v) as follows: j

For v ∈ V − Ci , let fi (v) = [1, 2]. j

j

j

For v ∈ Ci , fi (v) = [0, 1] if the jth bit in the binary representation of ni (v) is 1, else fi (v) = [2, 3]. Let Wi be the corresponding unit interval graph. j

Observation 8. For all i and j, the graph Wi is a supergraph of G. j

Proof. The only pairs of vertices in V (G) that are not adjacent in Wi are pairs contained in Ci , and hence they are also not adjacent in G.  Claim 9.

 1≤i≤⌈log k⌉;1≤j≤2b



j

Ui

j

Wi = G.

0≤i≤k−1;1≤j≤ti

Proof. In view of Observations 7 and 8, it suffices to show that if u and v are nonadjacent vertices in G, then u and v are j j nonadjacent in some Ui or some Wi . By construction, if φ(u) ̸= φ(v), then the binary expansions of the colors differ in j

j

some coordinate i, and then u and v are nonadjacent in Ui for some Ui used in the representation of Hi . By construction, if φ(u) = φ(v) = i, then the numbers assigned to u and v in the numbering of Ci differ in some coordinate j in their binary j expansion, and then u and v are nonadjacent in Wi .  From Claim 9 and Lemma 1, and noting that ti ≤ ⌈log α⌉ for 0 ≤ i ≤ k − 1, it immediately follows that cub(G) ≤ 2 ⌈log χ (G)⌉ box(G) + χ (G) ⌈log α(G)⌉. Tightness of Theorem 3. For ϵ > 0, we construct a graph G such that cub(G) ≤ 2 ⌈log χ (G)⌉ box(G) + χ (G) ⌈log α(G)⌉ ≤ cub(G)(1 + ϵ). Let k be a positive integer. Let Tk be the complete k-partite graph in which each part contains exactly nk

vertices. (Assume n to be a multiple of k.) We have box(Tk ) = k and cub(Tk ) = k log





for Tk equals 2k ⌈log k⌉ + k log 2+ϵ

 n k

n k



. The upper bound of Theorem 3

log k⌉ = cub(Tk )(1 + ⌈2⌈log n ) ≤ cub(Tk )(1 + ϵ), provided we take n sufficiently larger than k⌉

ϵ

k . Thus for all values of the boxicity there exist graphs such that the upper bound given by Theorem 3 is arbitrarily close in ratio to the true value of the cubicity. Acknowledgments The third author was supported by VATAT Post-doctoral Fellowship, Council of Higher Education, Israel and the Israel Science Foundation (grant number 862/10). References [1] Abhijin Adiga, Jasine Babu, L. Sunil Chandran, A constant factor approximation algorithm for boxicity of circular arc graphs, in: Algorithms and Data Structures Symposium (WADS), in: Lecture Notes in Computer Science, vol. 6844, Springer-Verlag, 2011. [2] Abhijin Adiga, L. Sunil Chandran, Naveen Sivadasan, Lower bounds for boxicity, Combinatorica 34 (6) (2014) 631–655.

446 [3] [4] [5] [6] [7]

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