Maximal sets of Hamilton cycles in Dn

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www.elsevier.com/locate/disc

Note

Maximal sets of Hamilton cycles in

Dn

Liqun Pu

a

, Hung-Lin Fu

b

, Hao Shen

c a_{Department of Mathematics, Zhengzhou University, Zhengzhou 450001, China} b_{Department of Applied Mathematics, National Chiao Tung University, Hsin Chu 30050, Taiwan}

c_{Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China} Received 4 December 2006; received in revised form 21 June 2007; accepted 5 July 2007

Available online 17 August 2007

Abstract

In this paper, we prove that there exists a maximal set of m directed Hamilton cycles inD_nif and only ifn/2mn − 1 for

n7.

Keywords: Directed Hamilton cycles; Complete directed graph; Maximal sets

1. Introduction

A Hamilton cycle in a graph T is a spanning cycle of T. If S is a set of edge-disjoint Hamilton cycles in T and ifE(S) is the set of edges occurring in the Hamilton cycles in S, then S is said to be maximal ifT − E(S) has no Hamilton cycle.

In paper[5], Hoffman et al. showed that there exists a maximal set S of m edge-disjoint Hamilton cycles in the complete graphK_nif and only if(n + 3)/4m(n − 1)/2. In paper[1], Bryant et al. showed that there exists a maximal set S of m edge-disjoint Hamilton cycles in the complete bipartite graphK_n,nif and only ifn/4 < mn/2. Later, Daven et al.[2]showed forn3 and p3, there exists a maximal set S of m Hamilton cycles in the complete multipartite graphK_np(p parts of size n) if and only if n(p − 1)/4mn(p − 1)/2, and m > n(p − 1)/4 if n is odd andp ≡ 1 (mod 4), except possibly if n is odd and m((n + 1)(p − 1) − 2)/4. Recently, Fu et al.[3]extended the results in[2]and proved that if(p − 1)/2mp − 1, then there exists a maximal set S of m Hamilton cycles inK2p− F , where F is a 1-factor of K2p.

In this paper, we will extend these results to directed graphs and study maximal sets of Hamilton cycles in the complete directed graphD_n.

Throughout this paper, we will use the following notation and terminology. A digraph D is said to be r-regular if deg+(v) = deg−(v) = r for each vertex v in D. Let D_nbe a complete directed graph of order n (without loops); it is clear thatD_nis(n − 1)-regular. For disjoint sets A and B, deﬁne the directed complete bipartite graph D_A,B as the

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LetV (Dm,n) be the vertex set of Dm,nand|V (Dm,n)| be the number of vertices in V (Dm,n). Let E(Dm,n) be the edge set ofDm,nand|E(Dm,n)| be the number of edges in E(Dm,n).

LetAi be an ordered i-set deﬁned onZ_t which can be expressed as slanted parentheses i.e.,Ai= (a1, a2, a3, . . . , ai−1, ai). Let A∗_i denote a path obtained fromAi,A∗_i = a1, a2, a3, . . . , ai−1, ai. Furthermore, for j ∈ Z_t, letAi+ j = (a1+ j, a2+ j, a3+ j, . . . , ai−1+ j, ai+ j) and A_i∗+ j = a1+ j, a2+ j, a3+ j, . . . , ai−1+ j, ai+ j. Note that all the entries inA_i + j and A∗_i + j are done modulo t.

2. Maximal sets of Hamilton cycles inD_n

Before proving the main result, we need the following lemma. Lemma 2.1. For an integer n,n7, Dnhas a Hamilton decomposition.

Proof. Dncan be regarded as twoKnby orienting properly. When n is odd,Knhas a Hamilton decomposition[7]. When n is even, Tillson[8]gives the proof.

The following lemma plays an important part to prove our main theorem.

Lemma 2.2. Let D be an r-regular directed graph on n vertices. Ifr n/2, then D contains a directed Hamilton

cycle.

In fact, this result is a special case of the following theorem which is obtained by Ghouila-Houri[4].

Theorem 2.1 (Ghouila-Houri[4]). Let +(D) and −(D) denote the minimum outdegree and minimum indegree in

D, respectively. If D is a strict digraph (without cycles of length 2), and min{+(D), −(D)}|V (D)|/2, then D is Hamiltonian.

On the other hand, ifD is regular strict digraph, then the condition for D to be Hamiltonian can be weaken slightly. The following result was obtained by Jackson.

Lemma 2.3 (Jackson[6]). Every strict digraph of minimum in-degree and out-degree k 2, on at most 2k +2 vertices,

is Hamiltonian.

Now, we are ready to prove our main result. In order to construct maximal sets of Hamilton cycles. By Lemma 2.3, we conclude that the minimum number m in a maximal set of m directed Hamilton cycles inD_nisn/2. Therefore, it sufﬁces to construct a maximal set of m directed Hamilton cycles for eachn/2mn − 1 for n7. In order to do that, the following lemma is essential. Mainly, we make an arrangement of directed differences in order that we can construct maximal sets of Hamilton cycles systematically.

Lemma 2.4. For positive integers l and m where m5 and 1l m − 1, there exists an ordered set A_m−l =

(a1, a2, . . . , am−l) ⊆ Zmsuch thatS = {(at+1− at) (mod m) | t = 1, 2, . . . , m − l − 1} is an (m − l − 1)-subset

of{1, 2, . . . , m − 1} (all directed differences are distinct).

Proof. We divide the proof into two cases.

Case 1: When m is even.

Whenl = 1, we delete the last element from the following ordered set (0, 1, m − 1, 2, m − 2, 3, m − 3, . . . , m/2 − 2, m/2 + 2, m/2 − 1, m/2 + 1, m/2) to get A_m−1, i.e., Am−1=0, 1, m − 1, 2, m − 2, 3, m − 3, . . . ,m 2 − 2, m 2 + 2, m 2 − 1, m 2 + 1 .

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1 3 0 4 1 2 0 6 5 7 5 2 3 4 Fig. 1. 1 3 0 5 1 3 5 0 7 2 4 2 4 6 8 6 Fig. 2.

Whenl = 2, we delete the last element in A_m−1and get Am−2 = 0, 1, m − 1, 2, m − 2, 3, m − 3, . . . ,m 2 − 2, m 2 + 2, m 2 − 1 . We give examples inFig. 1(i) (m = 6, l = 1) andFig. 1(ii) (m = 8, l = 1).

Note that inFig. 1(i),A5= (0, 1, 5, 2, 4), in which 3 is deleted and inFig. 1(ii),A7= (0, 1, 7, 2, 6, 3, 5), in which

4 is deleted.

We generalize the above method and get

Am−l= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0, 1, m − 1, 2, m − 2, 3, m − 3, 4, . . . ,m 2 − _l 2+ 2 ,m 2 + l 2 + 2 ,m 2 − l 2 + 1 ,m 2 + l 2+ 1 ,m 2 − l 2 if l ≡ 0 (mod 2); and 0, 1, m − 1, 2, m − 2, 3, m − 3, 4, . . . ,m 2 − l + 1 2 + 2 ,m 2 + l + 1 2 + 2 ,m 2 − l+1 2 + 1 ,m 2 + l + 1 2 + 1 ,m 2 −l + 1 2 , m 2 + l + 1 2 if l ≡ 1 (mod 2). Case 2: When m is odd.

Whenl = 1, we can get A_m−1fromZ_m\{(m + 1)/4}.

Am−1 = (0, 1, m − 1, 2, m − 2, 3, . . . , (m + 1)/4 − 1, m − (m + 1)/4 + 1, (m + 1)/4 + 1, m − (m + 1)/4, (m + 1)/4 + 2, m − (m + 1)/4 − 1, . . . , (m + 1)/2 + 2, (m + 1)/2 − 1, (m + 1)/2 + 1, (m + 1)/2).

Similarly, we can getA_m−2by deleting the last element inA_m−1. We give examples inFig. 2(i) (m = 7, l = 1) andFig. 2(ii) (m = 9, l = 1).

Note that inFig. 2(i),A6= (0, 1, 6, 3, 5, 4), in which 2 is deleted and inFig. 2(ii),A8= (0, 1, 8, 2, 7, 4, 6, 5), in

which 3 is deleted.

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Thus, we have Am−l= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0, 1, m − 1, 2, m − 2, 3, m − 3, 4, . . . , m + 1 4 − 1, m +1 − m + 1 4 , m + 1 4 + 1, m − m + 1 4 , m + 1 4 +2, m − _{m + 1} 4 − 1, . . . ,m + 1 2 − _l 2 + 1 ,m + 1 2 + l 2 + 1 ,m + 1 2 − l 2 ,m + 1 2 + l 2 if l ≡ 0 (mod 2); and 0, 1, m − 1, 2, m − 2, 3, m − 3, 4, . . . , m + 1 4 − 1, m + 1 − _{m + 1} 4 , _{m + 1} 4 + 1, m − _{m + 1} 4 , _{m + 1} 4 +2, m − m + 1 4 − 1, . . . ,m + 1 2 + l + 1 2 +1,m + 1 2 − l − 1 2 + 1 ,m + 1 2 + _{l + 1} 2 ,m + 1 2 − _{l − 1} 2 if l ≡ 1 (mod 2). With the above preparations, we are now in a position to prove the main theorem of this paper.

Theorem 2.2. For two positive integersm, n, n7, there exist maximal sets of m Hamilton cycles in Dnif and only if n/2mn − 1.

Proof. Suppose thatm < n/2, and let S be a set of m edge-disjoint directed Hamilton cycles in Dn. LetG(S) be the graph consisting of the union of the m Hamilton cycles in S. Then G(S)c is an r-regular directed graph with r = n − m − 1n/2 except the case n is odd, m = (n − 1)/2. So by Lemma 2.2, G(S)c_{contains a Hamilton cycle} and S is not a maximal set. When n is odd,m = (n − 1)/2, G(S)cis an(n − 1)/2-regular directed graph of order n, by Lemma 2.3,G(S)chas at least one Hamilton cycle. Thusmn/2.

Now we shall show whenn/2mn − 2, we can construct maximal sets of m Hamilton cycles in D_n.

When integerl > 0, let l = n − m − 1. Otherwise let integer l = 0. By Lemma 2.4, we can choose ordered set Am−landb_i (1i l) from Z_mand thusZ_mcan be partitioned intoA_m−land everything else not inA_m−l. That is, Zm= Am−l∪ {b1, b2, . . . , bl} where Am−l= (a1, a2, . . . , am−l) and Am−l∩ {b1, b2, . . . , bl} = ∅.

SinceD_n = D_n−m+ D_m,n−m+ D_m, let V (D_m,n−m) = {i0|i ∈ Z_n−m} ∪ {i|i ∈ Z_m}. Further, {i|i ∈ Z_m} can be partitioned asA_m−l∪ {b1, b2, . . . , b_l}. Now, we obtain a set S of directed Hamilton cycles with S = {(00, (b1+ j), 10, (b2+ j), . . . , (n − m − 2)0, (bn−m−1+ j), (n − m − 1)0, (A∗m−l+ j))|j ∈ Zm} from Dm+ Dm,n−m. Note that the entries of{bt+ j|1t n − m − 1} and A∗_m−l+ j are done modulo m. Obviously, E(Dm,n−m) ⊆ E(S) and Dn− E(S) is disconnected. Thus S is a maximal set of Hamilton cycles in Dnwhenn/2mn − 2.

As form = n − 1, Dncan be decomposed into directed Hamilton cycles by Lemma 2.1. Thus, we conclude the proof.

Example 1. Whenn = 8, 4m7, we give a construction of maximal sets of Hamilton cycles for m ∈ {4, 5, 6} and the casem = 7 can be settled by applying Lemma 2.1:

m = 4, the set is {(00, j, 10, 1 + j, 20, 2 + j, 30, 3 + j)|j ∈ Z4}. m = 5, the set is {(00, j, j + 1, j + 4, 10, 2 + j, 20, 3 + j)|j ∈ Z5}. m = 6, the set is {(00, j, j + 1, j + 5, j + 2, j + 4, 10, 3 + j)|j ∈ Z6}.

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Example 2. Whenn = 9, 5m8, we give a construction of maximal sets of Hamilton cycles for m ∈ {5, 6, 7} and the casem = 8 can be settled by applying Lemma 2.1:

m = 5, the set is {(00, j, j + 1, 10, j + 4, 20, 3 + j, 30, 2 + j)|j ∈ Z5}. m = 6, the set is {(00, j, j + 1, j + 5, j + 2, 10, 3 + j, 20, 4 + j)|j ∈ Z6}. m = 7, the set is {(00, j, j + 1, j + 6, j + 3, j + 5, j + 4, 10, 2 + j)|j ∈ Z7}. Acknowledgements

The authors wish to thank the referees for their valuable comments and suggestions, and in particular, one of the referees for his/her effort to provide assistance in rewriting the paper.

References

[1]D.E. Bryant, S. El-Zanati, C.A. Rodger, Maximal sets of Hamilton cycles inKn,n, J. Graph Theory 33 (2000) 25–31.

[2]M. Daven, J.A. MacDougall, C.A. Rodger, Maximal sets of Hamilton cycles in complete multipartite graphs, J. Graph Theory 43 (1) (2003) 49–66.

[3]H.L. Fu, S.L. Logan, C.A. Rodger, Maximal sets of Hamilton cycles inK2p− F , Discrete Math., to appear.

[4]A. Ghouila-Houri, Une condition sufﬁsante d’existence d’un circuit Hamiltonien, C. R. Acad. Sci. Paris 251 (1960) 495–497. [5]D.G. Hoffman, C.A. Rodger, A. Rosa, Maximal sets of 2-factors and Hamilton cycles, J. Combin. Theory (B) 57 (1993) 69–76. [6]B. Jackson, Long paths and cycles in oriented graphs, J. Graph Theory 5 (1981) 145–157.

[7]E. Lucas, Recreations Mathématiques, vol. 2, Gauthiers-Villars, Paris, 1892.