COM BIN A TOR I A L TOURNAMENTS THAT ADMIT EXACTLY ONE HAMILTONIAN CIRCUIT

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1 a l a 5 a a 2 a YEA R I February COM BIN A TOR I A L MAT HEM ATIC S TOURNAMENTS THAT ADMIT EXACTLY ONE HAMILTONIAN CIRCUIT by ROBERT J. DOUGLAS Departmet of Statistics Uiversity of North Carolia at Chapel Hill Istitute of Statistics Mimeo Series No JUNE 1969 Research partially sposored by the Air Force Office of Scietific Research, Office of Aerospace Research, Uited States Air Force uder Grat AFOSR-68-l406,

2 TOURNAMENTS THAT ADMIT EXACTLY ONE HAMILTONIAN CIRCUIT by Robert J. Douglas ABSTRACT I this paper is give a characterizatio of those touramets that admit exactly oe Hamiltoia circuit. I additio, we determie the umber of o-isomorphic touramets, with vertices, that admit a uique Hamiltoia circuit. I. INTRODUCTION A touramet is a directed graph where, betwee each two vertices v ad w, either the ordered pair (v, w) or (w, v) is a edge, but ot both. Touramets have bee studied by a umber of authors, ad Joh Moo's book [3] cotais a good review ad a extesive bibliography o the subject. The preset paper gives a solutio to a problem raised by Brako GrUbaum (oral commuicatio), amely fidig a characterizatio of those touramets that cotai a uique Hamiltoia circuit. (All paths ad circuits are simple ad directed, ad a path or circuit is Hamiltoia if it passes through all the vertices of the graph.) Frequetly i combiatorial mathematics, oce a characterizatio of a class of objects is foud, a eumeratio theorem follows. Such is the case here where we also give the exact umber of o-isomorphic touramets, with vertices, which admit a uique Hamiltoia circuit. Research partially sposored by the Air Force Office of Scietific Research, Office of Aerospace Research, Uited States Air Force uder Grat AFOSR

3 2 The aalogous problem for Hamiltoia paths is easy, ad is give by the followig theorem due to Redei [4]. THEOREM. The touramet T with vertices has a uique Hamil toia path vi' v 2 ', v if ad oly if T is orieted i the followig way: the directed edge (v., v.) 1. J is i T if ad oly if i < j. II. SOME NOTATION If S is ay set, IS I will be the cardiality of S. G: (V, E) will deote a graph G with vertex set V ad edge set E. T : (V, E) will deote a touramet with Ivi =. For v E V, p(v) [a(v)] is the umber of edges with v as iitial [termial] vertex. If P: v I'..., v k ad Q: wi'..., w m are paths i a graph, the P(v., vj), for i ::; j deotes the path vi' v i +I '..., v., ad 1. J P + Q deotes the path v I'..., vk ' w 2 '..., w m provided v = wi. k Fially, if G: (V, E) is a graph, ad V' -=- V, the S (V ') is the sectio graph determied by V', i.e., the graph with V' as vertex set ad whose edges are all those edges i E which coect two vertices i V'. III. STATEMENT OF THE RESULTS THEOREM 1. If C: vi'., v k ' v k + 1 ( = vi) is a (simple) circuit i a strogly coected touramet T : (V, E), the for ay vertex X E V ~ C, there exists a (simple) path P: wi'..., w m i V ~ C which has x as startig vertex or termial vertex, ad such that P ca be iserted betwee some two adjacet vertices of C to obtai

4 3 a loger circuit, Le., such that there exists a i where (vi' wi) E E ad (w m ' v i + I ) E E ad hece is a circuit. (P may cosist of oly the vertex x.) COROLLARY 1. I ay strogly coected touramet T, every (simple) circuit C ca be exteded to a Hamiltoia circuit H, i.e., there exists a Hamiltoia circuit H such that the order of the vertices of C i H is the same as their order i C. We thus obtai the followig result which abouds i the literature (cf. P. Camio [1], J.D. Foulkes [2], J.W. Moo [3]): COROLLARY 2. A touramet admits a Hamiltoia circuit if ad oly if it is strogly coected. PROPOSITION. For ~ 5, a touramet T : (V, E) admits a uique Hamiltoia circuit H if ad oly if the followig three coditios hold: There exist vertices C = {cl' " e, ck1 =I rp ad V = {d1 '..., d } ieossibly empty) ad a circuit H: x, c 1 ' m $ a, c k ' y, d,, c, d m' x I l I where ~ (1) = mtk+2 ad p (x) 1 a(y) I I I I I ad, for Ii - jl ~ 2, (c., c.) E E if i > j ~ J (d., d j ) E E if i < j ~ I I

5 4 ad (2) ad If i < j, r S s, ad the (c., d ) E E. J s (3) c C. / ~~''--'-... C. ~ J / H! / I I \ / x Y \ / / / ',I' / " ~,/ d '---._- u s ' d V r J H J Figure 1. Coditio 3 Figure 2. Notice that (1) completely determies the sectio graphs S({x, y, cl'..., c k }) ad S({x, y, d I, "., d m }), while (2) ad (3) are coditios o the edges joiig the sets C ad V. Also otice that (3) holds if ad oly if (3) holds for j = i + 1. COROLLARY 3. If T is strogly coected, ad p(v) ~ 2 for all v, or o(v) ~ 2 for all v, circuits. the T admits at least two Hamiltoia

6 5 We would like to replace the "forbidde subgraph" coditio(3) i the Propositio by a explicit descriptio of the orietatio o the edges with oe ed poit i C ad the other ed poit i V. To accomplish this,we ow defie (see figure 3) a class T of touramets T : (V, E) o ~ 5 vertices: DEFINITION OF T. Let C (possibly empty), V = {x, y} u C u V, ad let (1) ad (2) hold. If V = r/j, fix the T : (V, E) is completely described by (1). If V i= r/>, o ~ p ~ mi(m-1, k-1) 1 < i ~ k P (4) For j ~ i defie [c., c.l = {c.,..., c.}, 1. J 1. J lc., c.[ {c.+'..., c j }; similar defiitios are used for 1. J lc., c.], [d., d.], etc. If 51 c C ad 52':" V, defie 1. J 1. J to mea (sl' s2) E E for all sl E 51 ad s2 E 52' (A similar defiitio holds for "(52' 51) E E".) We will iductively describe all the edges betwee C ad V. We first give the edges with oe ed poit i [c l, c. l ad the other i V. 1. 0

7 6 Let (c., d. ) E E ~o J p ([ d 1 ' d. [, C. ) E E J p ~o (0, [C 1 ' C. [) E E ~o I I 1 I I Also let 1 => m of j. p So be ay subset of ]d., d ] J m p subject oly to the coditio that I r, I (5) ad let This starts the iductive defiitio. Now, for 1 ~ q ~ p, assume the edges with oe ed poit i [c 1 ' c i ] ad the other ed poit i 0 q-l are give. We will exted the defiitio to give the edges joiig the sets [c 1 ' c i ] ad O. To do this, we eed to give the edges betwee q ] c,c ] ad O. i i q-l q

8 7 For 1 ~ q ~ p, defie (] C., c. ], [d., d ]) E E 1 q-i l q J p _ q + 1 m ([d l, d. [, lc.,c. [) E E J p - q + 1 l q _ 1 l q (c.,d. ) E E l J _ q p q ( [dl' d. [, c ) E E. J i p _ q q Also let S be ay subset of q ad write ]d.,d. [ J p _ q J p _ q + 1 D q r i 1 (6) (c., S ) E E 1 q q (D "" S q' c. ) E E. q 1 q ) I Fially we let (] c i ' c k ], V) E E. P (7) REMARK. For q a,l,...,p, With ~ 5, each choice of the parameters k, p, i o ' "', i, 1 = jo' jl'..., j, SO'..., S (8) p p p defies, by meas of (1), (2), (4), (5), (6), ad (7), a touramet T : (V, E). Also it will be show (see the proof of Theorem 2 i Sectio IV) that ay two differet choices of parameters (8) result i isomorphically differet touramets. The collectio of these touramets is deoted by T

9 8 S([C p '1<]) k = 5 c. ~ q-l c. ~ p p(x) = 1 a (y) 1 d. Jo Figure 3.

10 9 THEOREM 2. (MAIN THEOREM) : A arbitrary touramet T, ~ 5 vertices, admits a uique Hamiltoia circuit H if ad oly if T is isomorphic to a elemet i T. choices for the parameters (8) defie isomorphica1ly differet touramets i T. o Furthermore, every two differet THEOREM 3. For ~ 5, there exist exactly l: k=l may o-isomorphic touramets T, o vertices, that admit a uique Hamiltoia circuit H. (Here if b > a.) IV. THE PROOFS OF THEOREM 1, THE PROPOSITION, THE REMARK, AND THEOREMS 2 AND 3. PROOF OF THEOREM 1: Let C: vi'..., v k ' vi be ay circuit T : (V, E), ad suppose x is a vertex of T ot i C. For each i, (vi' x) E E or (x, v.) E E. 1. Case 1: For each i = 1,..., k, (x, v. ) E E There exists a simple path Q i T from vi to x. Let v. be 1. the last vertex i Q which is also i C. The is a circuit i T which is loger tha C. (Here P w is the vertex i Q that follows v..) 1. Q(w, x) where

11 10 Case 2: Say (v., x) E E, ~ i = 1,.., k Reverse the orietatio o all the edges of T, agai reverse the orietatio o all the edges of apply Case 1, ad the T to obtai the required circuit. Case 3: Assume there exist p, q such that (v p' x) E E ad --- (x, v ) E E. q Without loss of geerality p = 1- Let i be the largest iteger such that (VI' x), (v 2 ' x),.., (Vi' x) are each i E. Hece 1 ~ i < q ad (x, v.+ ) E E. Cosequetly ~ 1 is a circuit loger tha C. (Here P cosists of just the vertex x.) This completes the proof of Theorem 1. DEFINITION. If H: v,..., v, 1 is a Hamiltoia circuit i a touramet T : (V, E) we defie v. to be a ~ c vertex (or a ~ c-vertex) if (v v) E E ad a tve a vertex (or a a-vertex) if i+i' i-i' ~ (Vi_I' v i + 1 ) E E. These defiitios deped o the give circuit H. [ IIc" is for the circuit IIa" is for " aticircuit".] See Figure 4. PROOF OF THE PROPOSITION: Assume T : (V, E) has exactly oe Hamiltoia circuit H:

12 11 c) V 2 H V 2 v. I J- ':'j+1 VI \ V i _ 1 /.-/ ~Vi+1 v. (type a) V 2 '_2 V2,_2 k:./ H 1 V 2 l / H Figure 4. Figure 5. Figure 6. LEMMA 1. H has at least oe c-vertex ad oe a-vertex. PROOF: If each v. is of type c~ ad 1 (see Figure 5) 2.-1 is odd~ the is a Hamiltoia circuit differet from H, a cotradictioo If each v. is of type c ad = 2. is eve~ the (see Figure 6) the circuit 1 ca be exteded~ by Corollaries 1 ad 2, to a Hamiltoia circuit differet from H. Now assume each v. is of type a~ = 2., ad see 1 Figure 7. If (V, V 0 ) E E ~ I 2-{..-2

13 12 the is aother Hamiltoia circuit; while if (v 0,v) E E, 2,(,.-2 1 the is a Hamiltoia circuit differet from H. type a, ad = 2-1, the Fially, if each v. ~ is of VI' v 3'..., v 2 -l' v2' v4'..., v2-2' v 1 is aother Hamiltoia circuit. (See Figure 8.) This fial cotradictio proves Lemma 1. H ) v 2-3 H ) V 2 _1 v 2-2 Figure 7. Figure 8.

14 13 Let t (v,) be the type (with respect to H) of the vertex v, ]. ]. LEMMA 2. If t(v, ) = c ad t(v i + 1 ) = a, the a(v,+ ) L ]. ]. 1 (See Figure 9.) Proof: Without loss of geerality i = 2. Say a(v ) > 3 l', so there is a j such that ~. j > 4 ad (v,, v 3 ) E E. But the (see J Figure 10) the circuit ca be exteded, by Corollaries 1 ad 2, to a Hamiltoia circuit differet from H, ad the cotradictio proves Lemma 2. v, J H Figure 9. Figure 10. As a cosequece of Lemmas 1 ad 2, we have that there exists exactly oe i such that t(v,) = c ad ]. t(v,+ ) = a. ]. 2 t(v,+ ) = a; ]. I furthermore, Without loss of geerality, we ca, therefore, say that the followig situatio holds:

15 14 For some k, 1 :0;; k :0;; -2, t(v.) = c for j = 1, k r (9) "l ", J t(v.) = a for j k+1, c,. J ) 1 I Sice t(v ) = a ad t(v] ) c, it follows from Lemma 2 that p (v ) 1. (10) To see this, reverse the orietatio o all the edges of Lemma 2, ad the agai reverse all orietatios. T, apply Therefore, usig (9), (10) ad Lemma 2, we ca write H i the followig way: where p(x) = 1 a(y) ad C = {c 1 '..., c k } +,p is the set of all the c-vertices, ad V u {x, y} is the set of all the a-vertices, where V = {d,, e " ", d } m (which may be empty). So k ~ 1 ad m + k + 2 =. (See Figure 2. ) H~+ J I x y y Figure 11. Figure 12.

16 15 Hece (2) holds, ad (1) will hold if for li-j I ~ 3 we ca prove that (c.,c.)ee 1 J for i > j (d., d.) E 1 J E for i < j. Assume there exist i ad j such that i - j ~ 3 ad (c j ' c i ) E E; see Figure 11. The is a circuit which ca be exteded to a Hamiltoia circuit differet from H, a cotradictio. (Here we assumed j > 1; if j 1, replace C. 1 by x i the above.) Now say there exist i ad j J- such that j - i ~ 3 ad (d., d. ) E E see Figure 12. The J 1 y, d., d. + H(d.,d. 1) + d. l' d j H(d j + 1,y) J 1 1 J- J-, is a circuit which ca be exteded to a Hamiltoia circuit differet from H. (If j m, replace d j + 1 by x i the above. ) This cotradictio proves (1) We ow will show that (3) holds. Say this is ot the case, i.e., there exist i < j ad r s s where (c., d ) E E, yet (d, c. ) E E. 1 r s J H ~ ci c 1 x Y x Y d ks d k 1 Figure 13. Figure 14.

17 16 The (see Figure 13) the circuit d + H(d,d ) + d, c. + H(c.,y) + y, x + H(x, c,) r rs s J J l. ca be exteded to aother Hamiltoia circuit; hece (3) is proved. To complete the proof of the Propositio, we must show that if a touramet T, o vertices, satisfies (1), (2), ad (3), the H is the oly Hamiltoia circuit i T. Assume H' is a Hamiltoia circuit differet from H. Edge (x, c 1 ) E H' as p(x) = 1. Let i ~ 1 be the largest iteger such that the path P: is a subpath of H', yet (c,, c i + 1 ) ~ H' (Such a i exists as if l., x, c 1 '..., c k is a path i H' the H H' as a(y) = 1 ad because there do ot exist r a s such that s > r ad (d, d ) E E.) Let (c i ' d ) E H' ad see Figure 14. By (3), there s r r does ot exist a j > i with (d, c,) E H'. As a(y) 1 ad r J because, by (1), there do ot exist u ad v such that v > u ad (d, d ) E E, v u P + cl." d, ~, (t,, d k ' c. r 1 1k2 s J must be a path i H' for some s where r < k < k <... < k ad 1 2 s j > i. But the we have (c., d ) E E ad (~, c, ) E E which col. r J s tradicts (3) This proves the Propositio. PROOF OF THE REMARK: Let A ad B [d,,d]. q q J _ p q m

18 17 Now (A, B ) E E by (7), so it suffices to show that (A, B ) E E P P q q implies (A B) E E Note that A c A ad B c B q-l' q-l' q - q-l q-l - q' whece (A, B ) E E, ad we are doe as (A - A,B ) E E q q-l q-l q q-l by (6). PROOF OF THEOREM 2: We otice that by the Propositio we are to show that a touramet belogs to T if ad oly if (1), (2), ad (3) are true. assume (1), (2), ad (3) hold for a touramet T, that T E T. We are doe if V First we ad we will show ~, so say V +~. Let i o be the smallest idex such that (c., d.) E E for some j, ad defie to 1 0 J as the smallest idex such that Certaily i o exists as (c k ' d 1 ) E E. Give i ad t, let i be the smallest q-l q-l q idex > i such that (c.,d. ) E E for some j < t q-l' ad defie q-j 1 q J t as the smallest idex < t such that q q-l Cotiue this procedure util we are forced to stop, say with q = 0, 1,..., p, defie i P ad t. P For t, q ad otice that jo = 1 as (ck,d 1 ) E E. Clearly (4) holds. (See Figure 3.) Now i o = 1 => jp +m as (c i ' d. ) E E ad (d, c ) E E. Defie J m 1 0 p

19 18 {d. : J $ m ad (c., d.) E E}, 1 0 J ad for q 1,., p, we defie s q {d. : J j p-q < j < J p- q+ ad (c ' d.) E E}, l i q J ad ote that as (d m, c l ) E E by (2). From the above defiitios, we immediately obtai (5), ad, except for (11) we also obtai (6). But so far we have ot used the assumptio that (3) holds for T. Usig (3), we coclude that (7) ad (11) hold, ad it therefore follows, usig the above parameters, that T E T We ext show that if T E T, the T satisfies (1), (2), ad (3). The oly thig to prove is that if V +, the (3) holds. Let ad (c., d ) E E. 1 r Case 1: For some q, 1 $ q $ p, c. E [c c. [ i q-l q By our defiitio of T, d E [d., d ). Now r $ s implies r J p- q+l d E [d., d ], ad i < j implies c. E ) c i c k ) Hece, by s J p- q+ l m J q-1 the Remark, (c., d) E E. J s m

20 19 Case 2: c. E [ C., c k ] P Thus c. E ] C., c k ], whece, by (7), (c., d ) E E. J 1. J s P Case 3: By defiitio of T, (D, [c 1 ' c. [) E E, 1. 0 whece (d, c.) E E, r 1. which shows that the hypothesis of (3) caot hold i Case 3. Sice these three cases exhaust all the possibilities, we have proved the first part of Theorem 1. Turig to the secod part of Theorem 2, we first ote that, by the defiitio of T, ( 1) ad ( 2) hold for ay T E T ' ad from (1) ad (2) we see that the collectio C i (1) is the set of all c-vertices i T (with respect to H) ad {x, y} u D is the set of a-vertices (with respect to H). Now let T I ad T 2 be isomorphic touramets i T. For i 1, 2, let Hi: i i i vi'..., v ' VI be the Hamiltoia circuit i T i give by (1) The image of HI uder the isomorphism must be H2 because T 2 has oly ~ Hamiltoia circuit. Without loss of geerality, we assume I v. ad v~ J J spod, for all j. The, for all J' v 1 is a c-vertex, j (with correrespect to Hl) if ad oly if is a c-vertex (with respect to H 2 ). Usig this, it easily follows that the correspodig parameters for ad T 2 are equal, which proves Theorem 2.

21 20 PROOF OF THEOREM 3: Fix ~ 5. By Theorem 2, we are to determie IT I. Defie ' T' [Til] as the set of all those touramets T E T such that V t r/j ad i O > l[i o = 1]. We first determie IT' I: We have k ~ 1, V ~~, ad m = - k - 2 ~ 1. As oted i Sectio III, the sectio graphs S({x, y, c 1 '.., ~}) ad S({x, y, d l,..., dm}) are uiquely determied by ad k. We have 1 < i < i <... < i ~ k 0 1 P 1 jo < j 1 <... < jp :::; -k-2 where o :::; p ~ mi(k-l, -k-3). Also, for q 1,..., p, S c ]d., d ] ad S c ]d., d. [ 0 - J p m q - J p _ q J p _ q + 1 Hece, for fixed k ad p, there exist (k-1) choices for the p+l sequece jo' jl'..., jp, -k-3..., i p ' ( P ) m-j 2 p choices for choices for the sequece So' ad 2jp-q+l-jp_q-l choices for each S q (q 1,..., p). Therefore we have the followig: IT'I -3 mi(k-l,-k-3) L L k=l p=o L: l=j < j < < j ~ -k-2 o 1 p 1 < i O < < i p :::; k <p(,k,p) (12)

22 21 where the ier sum, for fixed k ad p, is take over all sequeces i O ', i, such that the idicated equality ad P iequalities hold, ad where cjj(, k, p) m-j 2 P P IT q=l is the umber of choices for the sequece SO' Sl'.'0' Sp for a give..., j. p But reduces to m-jp 2 2jp-jo-P -k-p-3 2 ad hece is idepedet of the choice of jo'.. ', jp (ad of course..., i ). p Thus (12) reduces to IT'I -3 L k=1 mi(k-1,-k-3) L p=o 2- k- p- 3 (-k-3)(k-1) P P+1 We ow determie ITil I: If i = 1, the jp + m ad d <t SO' Hece So c ]d., d [, 0 m - J m p ad there exist 1 i < < i :s; k, ad o P 1 jo < < jp first case to obtai: m-j -1 2 p choices for choices for -k-4 ( P ) choices for < -k-2. With these chages, we proceed as i the -3 mi(k-1,-k-3) m-j -1 j -jo-p IT"I L L L 2 P. 2 P k=1 p=o l=j < <jp < -k-2 0 l=i < <i :s; k 0 P -3 mi(k-1,-k-3) -k-p-4 -k-4 = L l: 2 ( p ) (k-l) k=1 p=o P

23 have Fially, as there exists exactly oe T E: T with v, we 22 which proves Theorem 3. V REMARKS From Theorem 3, we obtai the followig table: T(), the umber of o-isomorphic touramets, with vertices, that admit a uique Hamiltoia circuit ~--1_3--:--:--:--2-:--5-:-l-4-:-3-~-: The three o-isomorphic touramets i T s : Figure 15.

24 23 The eight o-isomorphic touramets i T 5 : Figure 16.

25 24 REFERENCES 1. P. Cario, Chemis et Circuits Hami1toies des G~aphes Comp1ets, Comptes Redus, 249 (1959), J.D. Foulkes, Directed Graphs ad Assembly Schedules, Proc. Symp. Applied Math., Combiatorial Aalysis, 10 (1960), J.W. Moo, Topics o Touramets, Holt, Riehart ad Wisto, L. Redei, Ei kombiatorischer Satz. Acta Sci. Math. (Szeged), 7 (1934),