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Electronic Notes in Discrete Mathematics 44 (2013) 155–161 www.elsevier.com/locate/endm
Searching for perfection on hypergraphs Deborah Oliveros-Braniff 1,2 Instituto de Matem´ aticas, Unidad Juriquilla Universidad Nacional Aut´ onoma de M´exico
Natalia Garcia-Colin 3 Instituto de Matem´ aticas Universidad Nacional Aut´ onoma de M´exico
Amanda Montejano 4 UMDI-Facultad de Ciencias Universidad Nacional Aut´ onoma de M´exico
Luis Montejano 5 Instituto de Matem´ aticas, Unidad Juriquilla Universidad Nacional Aut´ onoma de M´exico
Abstract In this paper we study the family of oriented transitive 3–hypergraphs that arise from cyclic permutations and intervals in the circle, in order to search for the notion of perfection on hypergraphs. Keywords: transitive oriented 3–hypergraphs, cyclic permutations, perfection on hypergraphs.
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Introduction
We begin by proving a curious fact about permutations. Theorem 1.1 Let P be a permutation of [n] = {1, ..., n}. Then the minimum number of ascending subsequences of P that cover the whole sequence P is equal to the length of the largest descending subsequence of P . Proof. Define a directed graph D associated to the permutation P as follows. Let V (D) = [n] and, for 1 ≤ i < j ≤ n, the directed arc (i, j) from the vertex i to the vertex j, is in D if and only if the subsequence (i, j) is a descending subsequence of P . Note that D is actually a transitive oriented graph. Thus, the underlying graph G of D is a comparability graph, and so it is perfect. Hence χ(G) = ω(G). The statement follows since, by definition, there is a natural correspondence between the cliques of G and the maximal descending subsequences of P , as well as between the independent subsets of V (G) and the ascending subsequences of P . 2 The main idea in the previous proof is to associate to every permutation a transitive oriented graph whose underlying graph is perfect. (Related works can be found in the literature, but they approach the problem mostly from an algorithmic point of view, see [2,3,4]). We are interested in studying cyclic permutations. In such case, the purpose of the research is to associate to every cyclic permutation a 3–hypergraph with a natural structure of transitivity. As we will show, the 3–hypergraphs that arise from cyclic permutations are an excellent family to analyse within the problem of searching for perfection on hypergraphs.
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Cyclic permutations and transitive 3–hypergraphs
A cyclic permutation is a cyclic ordering of [n]. This is, an equivalence class [φ] of the set of bijections φ : [n] → [n], in respect to the cyclic equivalence relation. A cyclic permutation [φ] will be denoted by (φ(1) φ(2) . . . φ(n)). In resemblance to the transitive oriented graphs associated to linear permutations, we will now define the 3–hypergraphs associated to cyclic permu1 2 3 4 5
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tations and study some of its properties. We need first to introduce, as in [6], the notion of transitive orientation on a 3–hypergraph. Let H be a 3-hypergraph, that is, a pair of sets H = (V (H),E(H)) where V (H) V (H) is the vertex set of H, and the edge set of H is E(H) ⊆ 3 . Definition 2.1 Let H be a 3–hypergraph. An orientation of H is an assignment of exactly one of the two possible cyclic orderings to each edge. An orientation of a 3–hypergraph is called an oriented 3–hypergraph, and we will denote the oriented edges by O(H). Example 1 Let H = (V (H), E(H)) with V (H) = {a1 , a2 , a3 , a4 , a5 } and E(H) = {{a1 , a2 , a3 }, {a1 , a3 , a4 }, {a1 , a3 , a5 }} then a possible orientation of H could be O(H) = {(a1 , a2 , a3 ), (a1 , a4 , a3 ), (a1 , a3 , a5 )} obtaining an oriented 3–hypergraph. a2
a1
a5 a3
a4
Definition 2.2 An oriented 3–hypergraph H is transitive if whenever (u v z) and (z v w) ∈ O(H) then (u v w) ∈ O(H) (this implies (u w z) ∈ O(H)). Observe that in the example, since (a1 , a5 , a2 ) and (a2 , a5 , a3 ) are not in O(H) then H is not transitive. Let [φ] be a cyclic permutation. Three elements i, j, k ∈ [n], with i < j < k, are said to be in clock-wise order in respect to [φ] if there is ψ ∈ [φ] such that ψ −1 (i) < ψ −1 (j) < ψ −1 (k); otherwise the elements i, j, k are said to be in counter-clockwise order with respect to [φ]. Definition 2.3 The oriented 3–hypergraph H[φ] associated to a cyclic permutation [φ] is the hypergraph with vertex set V (H[φ] ) = [n], whose edges are the triplets {i, j, k}, with i < j < k, which are in clockwise order in respect to [φ], and whose edge orientations are induced by [φ]. An oriented 3–hypergraph H
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is called a cyclic permutation 3–hypergraph if there is a cyclic permutation [φ] such that H[φ] ∼ = H. It can be easily checked that, for any cyclic permutation [φ], its corresponding oriented 3–hypergraph H[φ] is transitive. Hence, in parallel to the study of perfection in transitive oriented graphs, using the following definitions of cromatic and clique numbers, the question below is very natural. For a cyclic permutation [φ], let the chromatic number χ([φ]) of [φ] be the minimum number of clock-wise subsequences of [φ] that cover the whole [φ], and the clique number ω([φ]) of [φ] be the length of the largest counterclockwise subsequence of [φ]. (1) It is true that χ([φ]) = w([φ])/2 for any cyclic permutation [φ]? This question is equivalent to asking if, for a cyclic permutation 3–hypergraph H, χ(H) = w(H)/2?, where ω(H) and χ(H) denote the clique number and the (weak) chromatic number of the unoriented hypergraph of H, respectively (see [5] for precise definitions). It is important to note that, in contrast to graphs where ω(G) ≤ χ(G), for 3–hypergraphs we have w(H)/2 ≤ χ(H). Another related question is: (2) It is true that any 3–hypergraph that can be transitively oriented, satisfies w(H)/2 = χ(H)? Unfortunately, as we will see later, both answers are negative. So, the following definition is a natural alternative. Definition 2.4 A 3–hypergraph H is perfect if every induced subhypergraph F of H, and every induced sub-hypergraph F of the complement of H, satisfy w(F )/2 = χ(F ). Now, the following question arises; (3) If both H and its complement can be transitively oriented, is it true that w(H)/2 = χ(H)? In the next section we will answer the latter question.
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Self–transitive 3–hypergraphs
An oriented 3–hypergraph containing all possible oriented 3–edges is called a 3–hypertournament. Let T Tn3 be the oriented 3–hypergraph associated to the cyclic permutation (1 2 . . . n). Clearly T Tn3 is a transitive 3–hypertournament.
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Theorem 3.1 There is just one transitive 3–hypertournament with n vertices, up to isomorphism. Theorem 3.1 allows us to hereafter refer to the transitive 3–hypertournament on n vertices. Although the proof is not difficult, we do not include it in this extended abstract. Besides, this fact is implied in the literature regarding cyclic orders [1]. For an oriented 3–hypergraph, it is not immediately clear how to define its oriented complement. However, for a spanning oriented subhypergraph of T Tn3 , H, we define its complement as the oriented 3–hypergraph H with V (H) = V (T Tn3 ) and O(H) = O(T Tn3 ) \ O(H). An oriented 3–hypergraph, H, such that it is a spanning subhypergraph of T Tn3 is called self-transitive if it is transitive and its complement is also transitive. In [6] the following theorem is proved. Theorem 3.2 H is an oriented cyclic permutation 3–hypergraph if and only if H is self transitive. Using Theorem 3.2 we show that not every self–transitive 3–hypergraph is perfect, by exhibiting a cyclic permutation whose corresponding oriented 3–hypergraph is not perfect. Let [φ] be the cyclic permutation of {1, 2, ..., 7} given by (1 6 4 7 3 2 5). The largest counter-clockwise subsequence of [φ] has length 4, but [φ] can not be decomposed into just 2 clockwise subsequences, hence showing that 2 = w(H[φ] )/2 < χ(H[φ] ) = 3. Nevertheless, we believe that a large number of cyclic permutation 3–hypergraphs are perfect. So, we propose the problem characterize perfect cyclic permutation 3–hypergraphs.
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Families of intervals in the circle
The intersection graph of a family of intervals in the line played an important role in the search for perfection in graphs. The same could also be possible for 3–hypergraphs. Let F be a finite family of closed intervals in the circle S1 , and let H = H(F) be the 3–hypergraph with V (H) = F such that a triple of intervals defines and edge of H if and only if the three intervals are pairwise disjoint. It is easy to see that H can be naturally associated to an oriented transitive 3–hypergraph. A sub-family of intervals I ⊂ F corresponds to a clique of H if and only if I is a pairwise disjoint subset of intervals of F. So, w(H) is the cardinality of the largest pairwise disjoint subset of intervals in F. On the other hand, a subset of intervals I ⊂ F corresponds to a (weak) independent set of vertices
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in H if and only if it satisfies the property (3, 2), this is, for any set of three intervals in I, at least two of them intersect. A (weak) coloring of H is an assignment of colors to the intervals of F such that each chromatic class corresponds to a (weak) independent subset of H, that is, a set satisfying the property (3, 2). Let χ(H) be the minimum ω(H) number of colors in a (weak) coloring of H so ≤ χ(H). 2 1 Theorem 4.1 Let F be a finite family of closed intervals in the circle S and = χ(H). let H be the 3–hypergraph associated to F. Then w(H) 2
Proof. We proceed by induction. If w(H) = 2, by definition, the family F satisfies the property (3, 2) which implies χ(H) = 1. Now, assume that w(H) = 2n + 1. Let {I0 , I1 , ..., I2n } ⊂ F be a pairwise disjoint set of intervals. Without loss of generality, we might assume that there is no other interval of F contained in I0 . Denote by G0 the set of intervals of F that intersect I0 , and note that it satisfies the property (3, 2). Thus, G0 corresponds to a independent set of H. On the other hand, F − G0 consists of all intervals contained in S1 − I0 , which may be regarded as the real line. The maximum number of pairwise disjoint intervals of F − G0 is 2n. Thus, we can pierce the intervals of F − G0 with 2n points, {v1 , ..., v2n } . For 1 ≤ i ≤ n, let Gi be the subset of intervals of F −G0 that intersects {v2i−1 , v2i }. Clearly, each Gi is an independent set because it satisfies the (3, 2) property. Consequently, χ(H) ≤ n + 1. Suppose now that w(H) = 2n. Let {I1 , I2 , ..., I2n } ⊂ F be a pairwise disjoint set of intervals. Without loss of generality, suppose that the distance between I1 and I2 is the smallest possible between all choices of pairwise disjoint subsets of F. This implies two facts; firstly, that the collection G0 of intervals of F which intersects {I1 , I2 } satisfies the property (3, 2) and, secondly that the collection of intervals F − G0 may be regarded as contained in the real line. The maximum number of pairwise disjoint intervals of F − G0 is 2n − 2, hence we can pierce the intervals of F − G0 with 2n − 2 points {v1 , ..., v2n−2 } . Let Gi be the subset of intervals of F − G0 that intersects {v2i−1 , v2i } for each i = 1, ...n − 1. Clearly, each Gi is independent because satisfies the property (3, 2). Consequently χ(H) ≤ n. This concludes the proof of the theorem. 2 In the search for perfection on 3–hypergraphs, it will now be important to characterize the families F of intervals in the circle for which the complement of H satisfies the property that w(H)/2 = χ(H).
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References [1] Peter Alles, Peter Jaroslav Nesetril and Svatopluk Poljak., Extendability, dimensions, and diagrams of cyclic orders, SIAM Journal on Discrete Mathematics 4 (1991) 453–471. [2] H. L. Bodlaender, T. Kloks, D. Kratsch Treewidth and pathwidth of permutation graphs, SIAM J. Comput., 8(4) (1995) 606-616. [3] S. Even, A. Pnueli, A. Lempel, Permutation and transitive graphs, Journal of the ACM, 19 3 (1972) 400-410. [4] J. Spinrad, On comparability and permutation graphs, SIAM J. Comput., 14(3) (1985) 658-670. [5] C. Berge, Hypergraph: combinatorics of finite sets, Elsevier Science Publishers B. V., (1989). [6] N. Garcia-Colin, A. Montejano, L. Montejano, D. Oliveros, Transitive oriented 3–Hypergraphs of cyclic orders, preprint.
