Cyclic Partitions of Complete Uniform Hypergraphs

By K n we denote the complete k-uniform hypergraph of order n, 1 6 k 6 n−1, i.e. the hypergraph with the set Vn = {1, 2, ..., n} of vertices and the set ( Vn k ) of edges. If there exists a permutation σ of the set Vn such that {E, σ(E), ..., σq−1(E)} is a partition of the set ( Vn k ) then we call it cyclic q-partition of K n and σ is said to be a (q, k)-complementing. In the paper, for arbitrary integers k, q and n, we give a necessary and sufficient condition for a permutation to be (q, k)-complementing permutation of K n . By K̃n we denote the hypergraph with the set of vertices Vn and the set of edges 2Vn − {∅, Vn}. If there is a permutation σ of Vn and a set E ⊂ 2Vn − {∅, Vn} such that {E, σ(E), ..., σp−1(E)} is a p-partition of 2Vn − {∅, Vn} then we call it a cyclic p-partition of Kn and we say that σ is p-complementing. We prove that K̃n has a cyclic p-partition if and only if p is prime and n is a power of p (and n > p). Moreover, any p-complementing permutation is cyclic. 1 Preliminaries and results Throughout the paper we will write Vn = {1, . . . , n}. For a set X we denote by ( X k ) the set of all k-subsets of X. A hypergraph H = (V ;E) is said to be k-uniform if E ⊂ ( V k ) (the cardinality of any edge is equal to k). We shall always assume that the set of vertices V of a hypergraph of order n is equal to Vn. The complete k-uniform hypergraph of order n is denoted by K (k) n , hence K (k) n = (Vn; ( Vn k ) ). Let σ be a permutation of the set Vn, let q be a positive integer, and let E ⊂ ( Vn k ) . If {E, σ(E), σ(E), . . . , σq−1(E)} is a partition of ( Vn k ) we call it a cyclic q-partition and σ is said to be (q, k)-complementing. It is ∗The research of APW was partially sponsored by polish Ministry of Science and Higher Education. the electronic journal of combinatorics 17 (2010), #R118 1 very easy to prove that then σ(E) = E. Write Ei = σ (E) for i = 0, ..., q − 1. It follows easily that σ(Ei) = Ei+t (mod q), for every integer t. If there is a cyclic 2-partition {E, σ(E)} of K n , we say that the hypergraph H = (Vn;E) is self-complementary and every (2, k)-complementing permutation of K (k) n is called self-complementing. In [16] we have given the characterization of selfcomplementing permutations which, as it turns out, is exactly Theorem 2 of this paper for p = 2, α = 1. Self-complementary k-uniform hypergraphs generalize the self-complementary graphs defined in [13] and [14]. The vertex transitive self-complementary k-uniform hypergraphs are the subject of the paper [11] by Potǒcnik and Šajna. Gosselin gave an algorithm to construct some special self-complementary k-uniform hypergraphs in [3]. In [6] and [10] Knor, Potǒcnik and Šajna study the existence of regular self-complementary k-uniform hypergraphs. The main result of this paper is a necessary and sufficient condition for a permutation σ of Vn to be (q, k)-complementing, where q is a positive integer (Theorem 3). In Theorem 5 we characterize integers n, k, α and primes p such that there exists a cyclic p-partition of K (k) n . Section 2 contains the proofs of Theorems 1, 2 and 3 given below. Section 3 is devoted to cyclic partitions of complete hypergraph K̃n = (Vn; 2 Vn − {∅, Vn}) (we call K̃n the general complete hypergraph of order n, to stress the distinction between complete uniform and complete hypergraphs). Theorem 1 Let n and k be integers, 0 < k < n, let p1 and p2 be two relatively prime integers. A permutation σ on the set Vn is (p1p2, k)-complementing if and only if σ is a (pj, k)-complementing for j = 1, 2. For integers n and d, d > 0, by r(n, d) we denote the reminder when n is divided by d. So we have n ≡ r(n, d) (mod d). For a positive integer k by Cp(k) we denote the maximum integer c such that k = p a, where a ∈ N (N stands for the sets of naturals, i.e. nonnegative integers). In other words, if k = ∑ i>0 kip , where 0 6 ki < p for every i ∈ {0, 1, . . .} (ki are digits with respect to basis p), then Cp(k) = min{i : ki 6= 0}. If A is a finite set, we write Cp(A) instead of Cp(|A|), for short. Theorem 2 Let n, p, k and α be positive integers, such that k < n and p is prime. A permutation σ of the set Vn with orbits O1, . . . , Om is (p , k)-complementing if and only if there is a non negative integer l such that the following two conditions hold: (i) r(n, p) < r(k, p), and (ii) ∑ i:Cp(Oi)l+α |Oi| ≡ 0 (mod p). Hence the condition (ii) of Theorem 2 could be written equivalently: ∑ i:Cp(Oi)0 nip i and k = ∑ i>0 kip i (0 6 ni, ki 6 p − 1 for every i). Cp( ( n k ) ) is equal to the number of indices i such that either ki > ni, or there exists an index j < i with kj > nj and kj+1 = nj+1, ..., ki = ni. Let p be a prime integer, 0 < k < n, k = ∑ i>0 kip , n = ∑ i>0 nip , where ki and ni are digits with respect to the basis p. Note that, by Theorem 2, if there is a cyclic p-partition of K (k) n then there are integers l and m, 0 6 m 6 l, such that nm < km, and nl+α−1 = nl+α−2 = ... = nl+1 = 0 (if α > 1), and ni = ki for m < i 6 l (if m < l). Conversely, if for indices l and m we have nl+α−1 = nl+α−2 = ... = nl+1 = 0 (for α > 1), nl = kl, nl−1 = kl−1, ..., nm+1 = km+1 (if m < l), and nm < km, then any permutation of Vn the electronic journal of combinatorics 17 (2010), #R118 3 which has two orbits O1 and O2 such that |O1| = ∑ i>l+α nip i and |O2| = ∑l+α−1 i=0 nip i = ∑l i=0 nip i is, by Theorem 2, (p, k)-complementing. We are thus led to the following corollary of Theorem 2. Theorem 5 Let n, k, p and α be positive integers such that k < n and p is prime. Suppose that k = ∑ i>0 kip , n = ∑ i>0 nip , where ki and ni are digits with respect to the basis p. The complete k-uniform hypergraph K (k) n has a cyclic p-partition if and only if there exist nonnegative integers l and m, m 6 l, such that nm < km, ni = ki for m < i 6 l, and nl+1 = nl+2 = ... = nl+α−1 = 0 (if α > 1). It is clear that for α > 1 it may happen that n, k and a prime p satisfy the assumption of Theorem 4, but violate the condition (i) of Theorem 2. Hence, in general, it is not true that if p divides ( n k ) then there is a cyclic p-partition of K (k) n . However, it is very easy to observe that Theorem 4 and Theorem 5 imply the following. Corollary 6 Let n, k and p be positive integers such that k < n and p is prime. The complete k-uniform hypergraph K (k) n has a cyclic p-partition if and only if p| ( n k ) . The problem whether for positive n, k and q there is a cyclic q-partition of K (k) n is in general open (unless q is a power of a prime).