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Vertex-disjoint cycles in bipartite tournaments
Discrete Mathematics 338 (2015) 1307–1309
Contents lists available at ScienceDirect
Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
Note
Vertex-disjoint cycles in bipartite tournaments✩ Yandong Bai a,b,∗ , Binlong Li a,d , Hao Li b,c a
Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an-710129, China
b
Laboratoire de Recherche en Informatique, C.N.R.S.-Université Paris-Sud, Orsay-91405, France
c
Institute for Interdisciplinary Research, Jiang Han University, Wuhan-430056, China
d
Department of Mathematics, University of West Bohemia, Pilsen-30614, Czech Republic
article
info
Article history: Received 23 June 2014 Received in revised form 8 January 2015 Accepted 20 February 2015
Keywords: Bipartite tournament Vertex-disjoint cycles Prescribed length Minimum outdegree Bermond–Thomassen conjecture
abstract Let t1 , . . . , tr ∈ [4, 2q] be any r even integers, where q ≥ 2 and r ≥ 1 are two integers. In this note, we show that every bipartite tournament with minimum outdegree at least qr − 1 contains r vertex-disjoint directed cycles of lengths t1′ , . . . , tr′ such that ti′ = ti for ti = 0 (mod 4) and ti′ ∈ {ti , ti + 2} for ti = 2 (mod 4), where 1 ≤ i ≤ r. The special case q = 2 of the result verifies the bipartite tournament case of a conjecture proposed by Bermond and Thomassen, stating that every digraph with minimum outdegree at least 2r − 1 contains at least r vertex-disjoint directed cycles. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Throughout the paper, a cycle (path) in a digraph always means a directed cycle (path). We use Bang-Jensen and Gutin [2] for terminology and notation not defined here. A digraph D = (V (D), E (D)) is strong if there exists a path from u to v for any two distinct vertices u and v of D. We − → write u → v and say v is an outneighbor of u if uv is an arc of D, write u → L if u → v for every v ∈ L and write L → u + if v → u for every v ∈ L. Define dD (v) = |{u : v → u and u ∈ V (D)}| and d− D (v) = |{u : u → v and u ∈ V (D)}| be the outdegree and indegree of v in D, respectively. Let δ + (D) and δ − (D) be the minimum outdegree and minimum indegree of D, respectively. Define a k-cycle to be a cycle of length k and a {k, l}-cycle to be a cycle of length either k or l. Two cycles are called vertex-disjoint if they share no common vertex. For a subdigraph S of D, let D[S ] and D − S be the subdigraphs of D induced on V (S ) and V (D) \ V (S ), respectively. A tournament is an orientation of a complete graph and a bipartite tournament is an orientation of a complete bipartite graph. In 2010, Lichiardopol [8] considered vertex-disjoint cycles of prescribed length in tournaments. The following result has been proved. Theorem 1.1 (Lichiardopol [8]). Let T be a tournament with min{δ + (T ), δ − (T )} ≥ (q − 1)r − 1. Then T contains r vertexdisjoint q-cycles. Lichiardopol [8] conjectured in the same paper that T contains r vertex-disjoint q-cycles if δ + (T ) ≥ (q − 1)r − 1. Motivated by the result and the conjecture above, we consider the analogous problem for bipartite tournaments, i.e., ✩ Supported by NSFC (Nos. 11171273, 11271300 and 11301371), CSC and NEXLIZ-CZ.1.07/2.3.00/30.0038.
∗
Corresponding author at: Laboratoire de Recherche en Informatique, C.N.R.S.-Université Paris-Sud, Orsay-91405, France. E-mail address: [email protected] (Y. Bai).
http://dx.doi.org/10.1016/j.disc.2015.02.012 0012-365X/© 2015 Elsevier B.V. All rights reserved.
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Y. Bai et al. / Discrete Mathematics 338 (2015) 1307–1309
Fig. 1. Regular bipartite tournament F4·k .
vertex-disjoint cycles of prescribed length in bipartite tournaments. Note that cycles in bipartite tournaments have been extensively studied in the literature, in particular, characterizations of bipartite tournaments having Hamilton cycle (Hamilton path, respectively) are known, see, e.g., [2] and [6]. In this note, the following results have been proved. Theorem 1.2. Let BT be a bipartite tournament with δ + (BT ) ≥ qr − 1. Then BT contains r vertex-disjoint cycles either of length 2q for even q or of lengths in {2q, 2q + 2} for odd q. Theorem 1.3. Let BT be a bipartite tournament with δ + (BT ) ≥ qr − 1 and let t1 , . . . , tr ∈ [4, 2q] be any r even integers. Then BT contains r vertex-disjoint cycles of length t1′ , . . . , tr′ such that ti′ = ti for ti = 0 (mod 4) and ti′ ∈ {ti , ti + 2} for ti = 2 (mod 4), where 1 ≤ i ≤ r. In 1981, Bermond and Thomassen [4] conjectured that every digraph with minimum outdegree at least 2r − 1 contains at least r vertex-disjoint cycles. This is trivially true for r = 1. Thomassen [10] in 1983 and Lichiardopol, Por and Sereni [9] in 2009 proved it for r = 2 and r = 3, respectively. In 2010, Bessy, Lichiardopol and Sereni [5] verified it for regular tournaments. In 2014, Bang-Jensen, Bessy and Thomassé [1] proved it for tournaments. Take q = 2 in Theorem 1.2, then the Bermond–Thomassen conjecture will be verified for bipartite tournaments. Corollary 1.1. Let BT be a bipartite tournament with δ + (BT ) ≥ 2r − 1. Then BT contains r vertex-disjoint 4-cycles. The proof of Theorem 1.3 will be presented at the end of this section and the proof of Theorem 1.2 will be presented in Section 2. We give some preliminary results as follows. Theorem 1.4 (Jackson [7]). Let BT be a strong bipartite tournament with δ + (BT ) ≥ s and δ − (BT ) ≥ t. Then BT contains a cycle of length at least 2(s + t ). Let F4·k = F (K , L, M , N ) be the k-regular bipartite tournament consisting of four independent sets K , L, M , N each of cardinality k, and with all possible arcs from K to L, from L to M, from M to N and from N to K (see Fig. 1). The following fact follows from the definition of F4·k . Fact 1.1. F4·k contains a 4k′ -cycle for all 1 ≤ k′ ≤ k. Theorem 1.5 (Beineke and Little [3]). Let C be a 2s-cycle of a bipartite tournament BT . If the sub-bipartite-tournament induced on C is not isomorphic to F4·k , where k = s/2, then BT contains a 2s′ -cycle for all 2 ≤ s′ ≤ s. Proof of Theorem 1.3. By Theorem 1.5 and Fact 1.1, the sub-bipartite-tournament induced on any 2q-cycle either contains a 2q′ -cycle for any 2 ≤ q′ ≤ q or contains a 2q′ -cycle for any even q′ with 2 ≤ q′ ≤ q. Then the result follows directly from Theorem 1.2. 2. Proof of Theorem 1.2 We can assume without loss of generality that BT is strong. In fact, if not, then we can choose a strong component with minimum outdegree at least qr − 1. Then δ − (BT ) ≥ 1 and BT has a cycle of length at least 2qr by Theorem 1.4. Thus BT has a 2q-cycle for even q or a (2q + 2)-cycle for odd q by Theorem 1.5 and Fact 1.1. We proceed by induction on r. It obviously holds for r = 1. Assume that r ≥ 2 and every bipartite tournament with minimum outdegree at least q(r − 1) − 1 contains r − 1 vertex-disjoint cycles either of length 2q for even q or of lengths in {2q, 2q + 2} for odd q. We distinguish two cases. Case 1. q is even. Note that BT has a 2q-cycle. Denote it by C and let BT ′ = BT − C . Then
δ + (BT ′ ) ≥ qr − 1 − q = q(r − 1) − 1. (1) By hypothesis BT ′ has r − 1 vertex-disjoint 2q-cycles. These cycles together with C form r vertex-disjoint 2q-cycles of BT .
Y. Bai et al. / Discrete Mathematics 338 (2015) 1307–1309
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Case 2. q is odd. If BT has a 2q-cycle C , then the same as Inequality (1) we have δ + (BT ′ ) ≥ q(r − 1)− 1 for BT ′ = BT − C . Thus by hypothesis BT ′ has r − 1 vertex-disjoint {2q, 2q + 2}-cycles. These cycles together with C form r vertex-disjoint {2q, 2q + 2}-cycles of BT . Now assume that BT has no 2q-cycle. We will show that BT has r vertex-disjoint (2q + 2)-cycles. Note that BT has a cycle of length at least 2qr. Let C be the maximum cycle of BT . Then |C | ≥ 2qr. Since BT contains no 2q-cycle, by Theorem 1.5, we have that C induces a F4·k for k = |C |/4. Recall that F4·k = F (K , L, M , N ) with |K | = |L| = |M | = |N | = k. For better presentation, we sometimes denote a path P from u to v by uP v in the following. If |C | ≥ r (2q + 2), then F4·k contains at least r vertex-disjoint (2q + 2)-cycles and thus the result holds. Now assume that |C | < r (2q + 2). We will get a contradiction by showing that BT has a cycle longer than C . Let (X , Y ) be a bipartition of BT with K , M ⊂ X and L, N ⊂ Y . Note that d+ C (v) =
|C | 4
<
r (q + 1)
(2)
2
for any v ∈ V (C ) and
δ + (BT ) ≥ qr − 1 >
r (q + 1) 2
> d+ C (v).
(3)
Thus X ∩ (BT − C ) ̸= ∅, Y ∩ (BT − C ) ̸= ∅ and every vertex of C has at least one outneighbor in BT − C . For any x ∈ X ∩ (BT − C ), assume without loss of generality that y1 → x for some y1 ∈ L and assume that there exists y2 ∈ L with x → y2 . Since |C | ≥ 2q + 2, then BT [C ] has a path P of length 2q − 2 from y2 to y1 . Now y2 Py1 xy2 is a 2q-cycle, a contradiction. By this contradiction and symmetry, we have the following:
• • • •
x → L or L → x for any x ∈ X ∩ (BT − C ); x → N or N → x for any x ∈ X ∩ (BT − C ); y → K or K → y for any y ∈ Y ∩ (BT − C ); y → M or M → y for any y ∈ Y ∩ (BT − C ).
For any y ∈ N, since d+ C (y) = k < qr − 1, there exists xK ∈ X ∩ (BT − C ) with N → xK . If L → xK , then since BT is strong there exists a path from xK to C . Assume that P = xK v1 . . . vp is a shortest one. Since N → xK and L → xK , we have v1 ̸∈ L ∪ N and p ≥ 2. If vp ∈ K ∪ M, then without loss of generality assume that vp ∈ K . Let P ′ be a Hamilton path of BT [C ] from vp to a vertex y′ ∈ N. Then xK P vp P ′ y′ xK is a cycle longer than C , a contradiction. If vp ∈ L ∪ N, then since L ∪ N → xK we have v1 ∈ Y ∩ (BT − C ) and v2 ∈ X ∩ (BT − C ). Thus p ≥ 3. Assume without loss of generality that vp ∈ L. Let x′′ , y′′ be two vertices in K and N respectively and let P ′′ be a Hamilton path of BT [C ] − {x′′ } from vp to y′′ . Then xK P vp P ′′ y′′ xK is a cycle longer than C , a contradiction. By this contradiction and symmetry, we have the following:
• • • •
xK → L and N → xK for some xK ∈ X ∩ (BT − C ); xM → N and L → xM for some xM ∈ X ∩ (BT − C ); yL → M and K → yL for some yL ∈ Y ∩ (BT − C ); yN → K and M → yN for some yN ∈ Y ∩ (BT − C ). Let x1 ∈ K , x2 , x′2 ∈ M, y1 , y′1 ∈ L and y2 ∈ N. Let P ∗ be a Hamilton path of BT [C ] − {x1 , x2 , y1 , y2 } from x′2 to y′1 . Then y′1 xM y2 xK y1 x2 yN x1 yL x′2 P ∗ y′1
(4)
is a cycle longer than C , a contradiction. The proof of Case 2 is complete. The proof of Theorem 1.2 is complete. References [1] J. Bang-Jensen, S. Bessy, S. Thomassé, Disjoint 3-cycles in tournaments: A proof of the Bermond–Thomassen conjecture for tournaments, J. Graph Theory 75 (3) (2014) 284–302. [2] J. Bang-Jensen, G. Gutin, Digraphs: Theory, Algorithms and Applications, Springer Verlag, London, 2008. [3] L.W. Beineke, C.H.C. Little, Cycles in bipartite tournaments, J. Combin. Theory Ser. B 32 (2) (1982) 140–145. [4] J.C. Bermond, C. Thomassen, Cycles in digraphs–a survey, J. Graph Theory 5 (1) (1981) 1–43. [5] S. Bessy, N. Lichiardopol, J.S. Sereni, Two proofs of the Bermond–Thomassen conjecture for tournaments with bounded minimum in-degree, Discrete Math. 310 (3) (2010) 557–560. [6] G. Gutin, Cycles and paths in semicomplete multipartite digraphs, theorems, and algorithms: a survey, J. Graph Theory 19 (4) (1995) 481–505. [7] B. Jackson, Long paths and cycles in oriented graphs, J. Graph Theory 5 (2) (1981) 145–157. [8] N. Lichiardopol, Vertex-disjoint directed cycles of prescribed length in tournaments with given minimum out-degree and in-degree, Discrete Math. 310 (19) (2010) 2567–2570. [9] N. Lichiardopol, A. Por, J.S. Sereni, A step toward the Bermond–Thomassen conjecture about disjoint cycles in digraphs, SIAM J. Discrete Math. 23 (2) (2009) 979–992. [10] C. Thomassen, Disjoint cycles in digraphs, Combinatorica 3 (3–4) (1983) 393–396.
Contents lists available at ScienceDirect
Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
Note
Vertex-disjoint cycles in bipartite tournaments✩ Yandong Bai a,b,∗ , Binlong Li a,d , Hao Li b,c a
Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an-710129, China
b
Laboratoire de Recherche en Informatique, C.N.R.S.-Université Paris-Sud, Orsay-91405, France
c
Institute for Interdisciplinary Research, Jiang Han University, Wuhan-430056, China
d
Department of Mathematics, University of West Bohemia, Pilsen-30614, Czech Republic
article
info
Article history: Received 23 June 2014 Received in revised form 8 January 2015 Accepted 20 February 2015
Keywords: Bipartite tournament Vertex-disjoint cycles Prescribed length Minimum outdegree Bermond–Thomassen conjecture
abstract Let t1 , . . . , tr ∈ [4, 2q] be any r even integers, where q ≥ 2 and r ≥ 1 are two integers. In this note, we show that every bipartite tournament with minimum outdegree at least qr − 1 contains r vertex-disjoint directed cycles of lengths t1′ , . . . , tr′ such that ti′ = ti for ti = 0 (mod 4) and ti′ ∈ {ti , ti + 2} for ti = 2 (mod 4), where 1 ≤ i ≤ r. The special case q = 2 of the result verifies the bipartite tournament case of a conjecture proposed by Bermond and Thomassen, stating that every digraph with minimum outdegree at least 2r − 1 contains at least r vertex-disjoint directed cycles. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Throughout the paper, a cycle (path) in a digraph always means a directed cycle (path). We use Bang-Jensen and Gutin [2] for terminology and notation not defined here. A digraph D = (V (D), E (D)) is strong if there exists a path from u to v for any two distinct vertices u and v of D. We − → write u → v and say v is an outneighbor of u if uv is an arc of D, write u → L if u → v for every v ∈ L and write L → u + if v → u for every v ∈ L. Define dD (v) = |{u : v → u and u ∈ V (D)}| and d− D (v) = |{u : u → v and u ∈ V (D)}| be the outdegree and indegree of v in D, respectively. Let δ + (D) and δ − (D) be the minimum outdegree and minimum indegree of D, respectively. Define a k-cycle to be a cycle of length k and a {k, l}-cycle to be a cycle of length either k or l. Two cycles are called vertex-disjoint if they share no common vertex. For a subdigraph S of D, let D[S ] and D − S be the subdigraphs of D induced on V (S ) and V (D) \ V (S ), respectively. A tournament is an orientation of a complete graph and a bipartite tournament is an orientation of a complete bipartite graph. In 2010, Lichiardopol [8] considered vertex-disjoint cycles of prescribed length in tournaments. The following result has been proved. Theorem 1.1 (Lichiardopol [8]). Let T be a tournament with min{δ + (T ), δ − (T )} ≥ (q − 1)r − 1. Then T contains r vertexdisjoint q-cycles. Lichiardopol [8] conjectured in the same paper that T contains r vertex-disjoint q-cycles if δ + (T ) ≥ (q − 1)r − 1. Motivated by the result and the conjecture above, we consider the analogous problem for bipartite tournaments, i.e., ✩ Supported by NSFC (Nos. 11171273, 11271300 and 11301371), CSC and NEXLIZ-CZ.1.07/2.3.00/30.0038.
∗
Corresponding author at: Laboratoire de Recherche en Informatique, C.N.R.S.-Université Paris-Sud, Orsay-91405, France. E-mail address: [email protected] (Y. Bai).
http://dx.doi.org/10.1016/j.disc.2015.02.012 0012-365X/© 2015 Elsevier B.V. All rights reserved.
1308
Y. Bai et al. / Discrete Mathematics 338 (2015) 1307–1309
Fig. 1. Regular bipartite tournament F4·k .
vertex-disjoint cycles of prescribed length in bipartite tournaments. Note that cycles in bipartite tournaments have been extensively studied in the literature, in particular, characterizations of bipartite tournaments having Hamilton cycle (Hamilton path, respectively) are known, see, e.g., [2] and [6]. In this note, the following results have been proved. Theorem 1.2. Let BT be a bipartite tournament with δ + (BT ) ≥ qr − 1. Then BT contains r vertex-disjoint cycles either of length 2q for even q or of lengths in {2q, 2q + 2} for odd q. Theorem 1.3. Let BT be a bipartite tournament with δ + (BT ) ≥ qr − 1 and let t1 , . . . , tr ∈ [4, 2q] be any r even integers. Then BT contains r vertex-disjoint cycles of length t1′ , . . . , tr′ such that ti′ = ti for ti = 0 (mod 4) and ti′ ∈ {ti , ti + 2} for ti = 2 (mod 4), where 1 ≤ i ≤ r. In 1981, Bermond and Thomassen [4] conjectured that every digraph with minimum outdegree at least 2r − 1 contains at least r vertex-disjoint cycles. This is trivially true for r = 1. Thomassen [10] in 1983 and Lichiardopol, Por and Sereni [9] in 2009 proved it for r = 2 and r = 3, respectively. In 2010, Bessy, Lichiardopol and Sereni [5] verified it for regular tournaments. In 2014, Bang-Jensen, Bessy and Thomassé [1] proved it for tournaments. Take q = 2 in Theorem 1.2, then the Bermond–Thomassen conjecture will be verified for bipartite tournaments. Corollary 1.1. Let BT be a bipartite tournament with δ + (BT ) ≥ 2r − 1. Then BT contains r vertex-disjoint 4-cycles. The proof of Theorem 1.3 will be presented at the end of this section and the proof of Theorem 1.2 will be presented in Section 2. We give some preliminary results as follows. Theorem 1.4 (Jackson [7]). Let BT be a strong bipartite tournament with δ + (BT ) ≥ s and δ − (BT ) ≥ t. Then BT contains a cycle of length at least 2(s + t ). Let F4·k = F (K , L, M , N ) be the k-regular bipartite tournament consisting of four independent sets K , L, M , N each of cardinality k, and with all possible arcs from K to L, from L to M, from M to N and from N to K (see Fig. 1). The following fact follows from the definition of F4·k . Fact 1.1. F4·k contains a 4k′ -cycle for all 1 ≤ k′ ≤ k. Theorem 1.5 (Beineke and Little [3]). Let C be a 2s-cycle of a bipartite tournament BT . If the sub-bipartite-tournament induced on C is not isomorphic to F4·k , where k = s/2, then BT contains a 2s′ -cycle for all 2 ≤ s′ ≤ s. Proof of Theorem 1.3. By Theorem 1.5 and Fact 1.1, the sub-bipartite-tournament induced on any 2q-cycle either contains a 2q′ -cycle for any 2 ≤ q′ ≤ q or contains a 2q′ -cycle for any even q′ with 2 ≤ q′ ≤ q. Then the result follows directly from Theorem 1.2. 2. Proof of Theorem 1.2 We can assume without loss of generality that BT is strong. In fact, if not, then we can choose a strong component with minimum outdegree at least qr − 1. Then δ − (BT ) ≥ 1 and BT has a cycle of length at least 2qr by Theorem 1.4. Thus BT has a 2q-cycle for even q or a (2q + 2)-cycle for odd q by Theorem 1.5 and Fact 1.1. We proceed by induction on r. It obviously holds for r = 1. Assume that r ≥ 2 and every bipartite tournament with minimum outdegree at least q(r − 1) − 1 contains r − 1 vertex-disjoint cycles either of length 2q for even q or of lengths in {2q, 2q + 2} for odd q. We distinguish two cases. Case 1. q is even. Note that BT has a 2q-cycle. Denote it by C and let BT ′ = BT − C . Then
δ + (BT ′ ) ≥ qr − 1 − q = q(r − 1) − 1. (1) By hypothesis BT ′ has r − 1 vertex-disjoint 2q-cycles. These cycles together with C form r vertex-disjoint 2q-cycles of BT .
Y. Bai et al. / Discrete Mathematics 338 (2015) 1307–1309
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Case 2. q is odd. If BT has a 2q-cycle C , then the same as Inequality (1) we have δ + (BT ′ ) ≥ q(r − 1)− 1 for BT ′ = BT − C . Thus by hypothesis BT ′ has r − 1 vertex-disjoint {2q, 2q + 2}-cycles. These cycles together with C form r vertex-disjoint {2q, 2q + 2}-cycles of BT . Now assume that BT has no 2q-cycle. We will show that BT has r vertex-disjoint (2q + 2)-cycles. Note that BT has a cycle of length at least 2qr. Let C be the maximum cycle of BT . Then |C | ≥ 2qr. Since BT contains no 2q-cycle, by Theorem 1.5, we have that C induces a F4·k for k = |C |/4. Recall that F4·k = F (K , L, M , N ) with |K | = |L| = |M | = |N | = k. For better presentation, we sometimes denote a path P from u to v by uP v in the following. If |C | ≥ r (2q + 2), then F4·k contains at least r vertex-disjoint (2q + 2)-cycles and thus the result holds. Now assume that |C | < r (2q + 2). We will get a contradiction by showing that BT has a cycle longer than C . Let (X , Y ) be a bipartition of BT with K , M ⊂ X and L, N ⊂ Y . Note that d+ C (v) =
|C | 4
<
r (q + 1)
(2)
2
for any v ∈ V (C ) and
δ + (BT ) ≥ qr − 1 >
r (q + 1) 2
> d+ C (v).
(3)
Thus X ∩ (BT − C ) ̸= ∅, Y ∩ (BT − C ) ̸= ∅ and every vertex of C has at least one outneighbor in BT − C . For any x ∈ X ∩ (BT − C ), assume without loss of generality that y1 → x for some y1 ∈ L and assume that there exists y2 ∈ L with x → y2 . Since |C | ≥ 2q + 2, then BT [C ] has a path P of length 2q − 2 from y2 to y1 . Now y2 Py1 xy2 is a 2q-cycle, a contradiction. By this contradiction and symmetry, we have the following:
• • • •
x → L or L → x for any x ∈ X ∩ (BT − C ); x → N or N → x for any x ∈ X ∩ (BT − C ); y → K or K → y for any y ∈ Y ∩ (BT − C ); y → M or M → y for any y ∈ Y ∩ (BT − C ).
For any y ∈ N, since d+ C (y) = k < qr − 1, there exists xK ∈ X ∩ (BT − C ) with N → xK . If L → xK , then since BT is strong there exists a path from xK to C . Assume that P = xK v1 . . . vp is a shortest one. Since N → xK and L → xK , we have v1 ̸∈ L ∪ N and p ≥ 2. If vp ∈ K ∪ M, then without loss of generality assume that vp ∈ K . Let P ′ be a Hamilton path of BT [C ] from vp to a vertex y′ ∈ N. Then xK P vp P ′ y′ xK is a cycle longer than C , a contradiction. If vp ∈ L ∪ N, then since L ∪ N → xK we have v1 ∈ Y ∩ (BT − C ) and v2 ∈ X ∩ (BT − C ). Thus p ≥ 3. Assume without loss of generality that vp ∈ L. Let x′′ , y′′ be two vertices in K and N respectively and let P ′′ be a Hamilton path of BT [C ] − {x′′ } from vp to y′′ . Then xK P vp P ′′ y′′ xK is a cycle longer than C , a contradiction. By this contradiction and symmetry, we have the following:
• • • •
xK → L and N → xK for some xK ∈ X ∩ (BT − C ); xM → N and L → xM for some xM ∈ X ∩ (BT − C ); yL → M and K → yL for some yL ∈ Y ∩ (BT − C ); yN → K and M → yN for some yN ∈ Y ∩ (BT − C ). Let x1 ∈ K , x2 , x′2 ∈ M, y1 , y′1 ∈ L and y2 ∈ N. Let P ∗ be a Hamilton path of BT [C ] − {x1 , x2 , y1 , y2 } from x′2 to y′1 . Then y′1 xM y2 xK y1 x2 yN x1 yL x′2 P ∗ y′1
(4)
is a cycle longer than C , a contradiction. The proof of Case 2 is complete. The proof of Theorem 1.2 is complete. References [1] J. Bang-Jensen, S. Bessy, S. Thomassé, Disjoint 3-cycles in tournaments: A proof of the Bermond–Thomassen conjecture for tournaments, J. Graph Theory 75 (3) (2014) 284–302. [2] J. Bang-Jensen, G. Gutin, Digraphs: Theory, Algorithms and Applications, Springer Verlag, London, 2008. [3] L.W. Beineke, C.H.C. Little, Cycles in bipartite tournaments, J. Combin. Theory Ser. B 32 (2) (1982) 140–145. [4] J.C. Bermond, C. Thomassen, Cycles in digraphs–a survey, J. Graph Theory 5 (1) (1981) 1–43. [5] S. Bessy, N. Lichiardopol, J.S. Sereni, Two proofs of the Bermond–Thomassen conjecture for tournaments with bounded minimum in-degree, Discrete Math. 310 (3) (2010) 557–560. [6] G. Gutin, Cycles and paths in semicomplete multipartite digraphs, theorems, and algorithms: a survey, J. Graph Theory 19 (4) (1995) 481–505. [7] B. Jackson, Long paths and cycles in oriented graphs, J. Graph Theory 5 (2) (1981) 145–157. [8] N. Lichiardopol, Vertex-disjoint directed cycles of prescribed length in tournaments with given minimum out-degree and in-degree, Discrete Math. 310 (19) (2010) 2567–2570. [9] N. Lichiardopol, A. Por, J.S. Sereni, A step toward the Bermond–Thomassen conjecture about disjoint cycles in digraphs, SIAM J. Discrete Math. 23 (2) (2009) 979–992. [10] C. Thomassen, Disjoint cycles in digraphs, Combinatorica 3 (3–4) (1983) 393–396.