Self-complementing permutations of k-uniform hypergraphs

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1 Dscrete Mathematcs Theoretcal Computer Scence DMTCS vol. 11:1, 2009, Self-complementng permutatons of k-unform hypergraphs Artur Szymańsk A. Paweł Wojda Faculty of Appled Mathematcs, AGH Unversty of Scence Technology, Al. Mckewcza 30, Kraków, Pol receved January 20, 2008, revsed January 16, 2009, accepted January 23, A k-unform hypergraph H = (V ; E) s sad to be self-complementary whenever t s somorphc wth ts complement H = (V ; `V k E). Every permutaton σ of the set V such that σ(e) s an edge of H f only f e E s called selfcomplementng. 2-self-comlementary hypergraphs are exactly self complementary graphs ntroduced ndependently by Rngel (1963) Sachs (1962). For any postve nteger n we denote by λ(n) the unque nteger such that n = 2 λ(n) c, where c s odd. In the paper we prove that a permutaton σ of [1, n] wth orbts O 1,..., O m s a self-complementng permutaton of a k-unform hypergraph of order n f only f there s an nteger l 0 such that k = a2 l + s, a s odd, 0 s < 2 l the followng two condtons hold: () n = b2 l+1 + r, r {0,..., 2 l 1 + s}, () P :λ( O ) l O r. For k = 2 ths result s the very well known characterzaton of self-complementng permutaton of graphs gven by Rngel Sachs. Keywords: Self-complementng permutatons, k-unform hypergraphs 1 Introducton Let V be a set of n elements. The set of all k-subsets of V s denoted by ( V. A k-unform hypergraph H conssts of a vertex-set V (H) an edge-set E(H) ( ) V (H) k. Two k-unform hypergraphs G H are somorphc, f there s a bjecton σ : V (G) V (H) such that e E(G) f only f {σ(x) x e} E(H). The complement of a k- unform hypergraph H s the hypergraph H such that V (H) = V (H) the edge set of whch conssts of all k-subsets of V (H) not n E(H) (n other words E(H) = ( ) V (H) k E). A k-unform hypergraph H s called self-complementary (s-c for short) f t s somorphc wth ts complement H. Isomorphsm of a k-unform self-complementary hypergraph onto ts complement s called self-complementng permutaton (or s-c permutaton) c 2009 Dscrete Mathematcs Theoretcal Computer Scence (DMTCS), Nancy, France

2 118 Artur Szymańsk A. Paweł Wojda The 2-unform self-complementary hypergraphs are exactly self-complementary graphs. Ths class of graphs has been ndependently dscovered by Rngel Sachs who proved the followng. Theorem 1 (Rngel (Rn63) Sachs (Sac62)) Let n be a postve nteger. A permutaton σ of [1, n] s a self-complementng permutaton of a self-complementary graph of order n f only f all the orbts of σ have ther cardnaltes congruent to 0 (mod 4) except, possbly, one orbt of cardnalty 1. Observe that by Theorem 1 an s-c graph of order n exsts f only f n 0 or n 1 (mod 4) or, equvalently, whenever ( n 2) s even. In (SW) we prove that a smlar result s true for k-unform hypergraphs. Theorem 2 ((SW)) Let k n be postve ntegers, k n. A k-unform self-complementary hypergraph of order n exsts f only f ( n s even. A smple crteron for evenness of ( n has been gven n (Gla99) ( then redscovered n (KHRM58)). Theorem 3 ((Gla99; KHRM58)) Let k n be postve ntegers, k = + =0 c 2 n = + =0 d 2, where c, d {0, 1} for every. ( n s even f only f there s 0 such that c 0 = 1 d 0 = 0. Theorem 3 asserts that ( n s even f only f k has 1 n a certan bnary place whle n has 0 n the correspondng bnary place. For example, ( 27 13) s even snce 13 = = (so we have c 2 = 1 d 2 = 0). Except for Theorem 1 whch s a characterzaton of the self-complementng permutatons for graphs, there are already two publshed results characterzng the permutatons of k-unform s-c hypergraphs for k > 2. Namely, Kocay n (Koc92) (see also (Pal73)) Szymańsk n (Szy05) have characterzed the s-c permutatons of s-c k-unform hypergraphs for, respectvely, k = 3 k = 4. Ths work s a contnuaton of the work of (SW) (Woj06). We generalze all the results mentoned above by gvng a characterzaton of the s-c permutatons of k-unform hypergraphs for any ntegers k n. 2 Result Any postve nteger n may be wrten n the form n = 2 l c, where c s an odd nteger. Moreover, l c are unquely determned. We wrte then λ(n) = l. Note that n the bnary expanson of n, λ(n) s the ndex of the frst 1 bt. For any set A we shall wrte λ(a) n place of λ( A ), for short. In the proof of our man result we shall need the followng lemma proved n (Woj06). Lemma 1 Let k, m n be postve ntegers, let σ : V V be a permutaton of a set V, V = n, wth orbts O 1,..., O m. σ s a self-complementng permutaton of a self-complementary k-unform hypergraph, f only f, for every p {1,..., k} for every decomposton k = k k p of k (k j > 0 for j = 1,..., p), for every subsequence of orbts O 1,..., O p such that k j O j for j = 1,..., p, there s a subscrpt j 0 {1,..., p} such that λ(k j0 ) < λ(o j0 )

3 Self-complementng permutatons of k-unform hypergraphs 119 Gven any nteger l 0. If the bnary expanson of k s 1 bt n poston l, then k can be wrtten n the form k = a l 2 l + s l, where a l s odd 0 s l < 2 l. Theorem 4 Let k n be ntegers, k n. A permutaton σ of [1, n] wth orbts O 1,..., O m s a selfcomplementng permutaton of a k-unform hypergraph of order n f only f there s a nonnegatve nteger l such that k = a l 2 l + s l, where a l s odd 0 s l < 2 l, the followng two condtons hold: () n = b l 2 l+1 + r l, r l {0,..., 2 l 1 + s l }, () :λ(o ) l O r l. Proof: Suffcency. By contradcton. Let n, k, l, a l, b l, s l r l be ntegers verfyng the condtons of the theorem, let σ be a permutaton of [1, n] wth orbts O 1,..., O m verfyng (), let us suppose that σ s not a s-c permutaton of any k-unform s-c hypergraph of order n. Then, by Lemma 1, there s a decomposton of k = k k t a subsequence of orbts O 1,..., O t such that 0 < k j O j (1) λ(k j ) λ(o j ) (2) for j = 1,..., t. Snce a l s odd, we have k 2 l + s l (mod 2 l+1 ). By (2), j: λ(o j )>l k j 0 (mod 2 l+1 ). Therefore k = t k j = j=1 j: λ(o j )>l k j + j: λ(o j ) l k j j: λ(o j ) l k j (mod 2 l+1 ) Hence, by (1), () () we have j:λ(o j ) l k j j:λ(o j ) l O j < 2 l+1, therefore a contradcton. 2 l + s l = j: λ(o j ) l k j j: λ(o j ) l O j r l < 2 l + s l Necessty. Let 1 k n let σ be a permutaton of the set [1, n] wth orbts O 1,..., O m. Let us suppose that for every nteger l such that k = a l 2 l + s l, where a l s odd postve nteger, 0 s l < 2 l, n = b l 2 l+1 + r l, 0 r l < 2 l+1 we have ether or r l {2 l + s l,..., 2 l+1 1} r l {0,..., 2 l 1 + s l } : λ(o ) l O > r l We shall prove that σ s not a s-c permutaton of any s-c k-unform hypergraph of order n. For ths purpose we shall gve two clams.

4 120 Artur Szymańsk A. Paweł Wojda Clam 1 For every nonnegatve nteger l such that k = a l 2 l + s l, where a l s odd 0 s l < 2 l, we have O 2 l + s l Proof of Clam 1. : λ(o ) l Let us wrte :λ(o ) l O :λ(o )>l O n ther bnary forms: :λ(o ) l :λ(o )>l O = O = e j 2 j j=0 f j 2 j where e j, f j {0, 1} for every j. Observe that f j = 0 for j = 0,..., l therefore We shall consder two cases. j=0 l e j 2 j = r l (3) Case 1. r l {0,..., 2 l + s l 1} :λ(o O ) l > r l. We have n 2 l+1 (otherwse r l = n = :λ(o O ) l ). Snce j=0 e j2 j > r l, by (3), we obtan j=0 e j2 j 2 l+1 > 2 l + s l. j=0 Case 2. r l {2 l + s l,..., 2 l+1 1}. We have :λ(o ) l O = j=0 e j2 j l j=0 e j2 j = r l 2 l + s l, the clam s proved. Clam 2 Let α 1,..., α q λ 1,..., λ q be ntegers such that 0 < α, 0 λ λ(α ) λ l for = 1,..., q q =1 α 2 l. Then there are β 1,..., β q such that for every = 1,..., q 0 β α (4) ether β = 0 or λ(β ) λ (5) q β = 2 l (6) =1 Proof of Clam 2. The exstence of β 1,..., β q verfyng (4)-(5) q =1 β 2 l s very easy. Indeed, t s mmedate that β 1 = 2 λ1, β 2 =... β q = 0 s a sequence wth the desred propertes. So let us suppose that β 1,..., β q s a sequence verfyng (4)-(5) q =1 β 2 l such that q =1 β s maxmal. If q =1 β = 2 l then the proof s complete. So let us suppose that q =1 β < 2 l. Then there s 0 {1,..., q} such that β 0 < α 0. Observe that β λ 0 α 0. The sequence β 1,..., β q defned by β 0 = β λ 0 β = β for 0 also verfes (4)-(5) q =1 β 2 l, whch contradcts the maxmalty of the sum q =1 β, the clam s proved. We shall use our clams to construct a decomposton of k n the form k = k k m such that

5 Self-complementng permutatons of k-unform hypergraphs 121 (1) k 1,..., k m are nonnegatve ntegers, (2) k O for = 1,..., m, (3) λ(k ) λ(o ) whenever k > 0 By Lemma 1, ths wll mply that σ s not a s-c permutaton of any k-unform s-c hypergraph. Let us wrte k n ts bnary form: k = 2 lt + 2 lt l1 + 2 l0 where l 0 < l 1 <... < l t. By Clam 1, :λ(o ) l 0 O 2 l0. Hence, by Clam 2, there are nonnegatve ntegers k (0) 1, k(0) 2,..., k(0) m such that k (0) = 0 for such that λ(o ) > l 0 k (0) O for = 1,..., m λ(k (0) ) λ(o ) whenever k (0) > 0 m =1 k (0) = 2 l0 Note that, for = 1,..., m, we have λ( O k (0) ) λ(o ). Let us suppose that we have already constructed k (j) 1,..., k(j) m, (j t), such that k (j) = 0 for such that λ(0 ) > l O for = 1,..., m k (j) λ(k (j) m =0 ) λ(o ) whenever k (j) > 0 k (j) = 2 lj + 2 lj l0 λ( O k (j) ) λ(o ) If j = t, then we have already found a desred decomposton of k. If j < t, then, by Clam 1, we have :λ(o ) l j+1 ( O k (j) ) 2 lj+1. λ( O k (j) ) λ(o ) for every {1,..., m} such that O k (j) > 0. Hence, by Clam 2, there are β 1,..., β m such that β = 0 for such that λ(o ) > l j+1 0 β O k (j) for = 1,..., m λ(o ) λ(β ) for = 1,..., m whenever β 0 m β = 2 lj+1 =1

6 122 Artur Szymańsk A. Paweł Wojda Thus we may defne for every = 1,..., m k (j+1) = k (j) + β to obtan the sequence (k (j+1) 1,..., k (j+1) m ) verfyng for every {1,..., m} k (j+1) = 0 for such that λ(o ) > l j+1 k (j+1) O λ(k (j+1) ) λ(o ) whenever k (j+1) > 0 m =1 k (j+1) = 2 lj lj l0 It s clear that k = m =1 k(t) the proof of Theorem 4 s complete. Theorem 4 mples very easly the followng theorem frst proved by Kocay. Corollary 1 (Kocay (Koc92)) σ s a self-complementng permutaton of a self-complementary 3-unform hypergraph f only f ether all the orbts of σ have even cardnaltes, or else, t has 1 or 2 fxed ponts the all remanng orbts of σ have ther cardnaltes beng multples of 4. For k = 2 l Theorem 4 may be wrtten as follows. Corollary 2 Let l n be nonnegatve ntegers, 2 l < n, let 0 r < 2 l+1 be such that n r (mod 2 l+1 ). A permutaton σ of [1, n] wth orbts O 1,... O m s a self-complementng permutaton of a 2 l -unform self-complementary hypergraph f only f () r {0,..., 2 l 1} () :λ(o ) l O r. Theorem 2 for l = 1 (.e. for graphs) s exactly Theorem 1, for l = 2 the followng theorem proved by Szymańsk n (Szy05). Corollary 3 A permutaton σ s self-complementng permutaton of a 4-unform hypergraph of order n f only f n r( mod 8) wth r = 0, 1, 2 or 3, the sum of the cardnaltes of orbts whch are not multples of 8 s at most 3. Acknowledgements The research was partally supported by AGH local grant No

7 Self-complementng permutatons of k-unform hypergraphs 123 References [Gla99] J.W.L. Glasher. On the resdue of a bnomal coeffcent wth respect to a prme modulus. Quarterly Journal of Mathematcs, 30: , [KHRM58] S.H. Kmball, T.R. Hatcher, J.A. Rley, L. Moser. Soluton to problem e1288: Odd bnomal coeffcents. Amer. Math. Monthly, 65: , [Koc92] W. Kocay. Reconstructng graphs as subsumed graphs of hypergraphs, some selfcomplementary trple systems. Graphs Combnatorcs, 8: , [Pal73] E.M. Palmer. On the number of n-plexes. Dscrete Math., 6: , [Rn63] G. Rngel. Selbstkomplementare graphen. Arch. Math., 14: , [Sac62] H. Sachs. Uber selbstkomplementare graphen. Publ. Math. Debrecen, 9: , [SW] A. Szymańsk A.P. Wojda. A note on k-unform self-complementary hypergraphs of gven order. submted. [Szy05] A. Szymańsk. A note on self-complementary 4-unform hypergraphs. Opuscula Mathematca, 25/2: , [Woj06] A.P. Wojda. Self-complementary hypergraphs. Dscussones Mathematcae Graph Theory, 26: , 2006.

8 124 Artur Szymańsk A. Paweł Wojda