Hamiltonian properties of Toeplitz graphs - ScienceDirect
DISCRETE MATHEMATICS ELSEVIER
Discrete Mathematics 159 (1996) 69-81
Hamiltonian properties of Toeplitz graphs Ren6 van D a P '1, G e r t Tijssen a, Zsolt T u z a b, J a c k A.A. v a n der Veen c, C h r i s t i n a Z a m f i r e s c u d, T u d o r Z a m f i r e s c u e'* aDepartment of Econometrics, University of Groningen, P.O. Box 800, 9700 A V Groningen, Netherlands bComputer and Automation Institute, Hungarian Academy of Sciences, H-1111 Budapest, Kende u. 13-17, Hungary CDepartment of Quantitative Methods, Nijenrode, Netherlands School of Business, Straatweg 25, 3621 BG Breukelen, Netherlands aDepartment of Computer Science, Hunter College, City University of New York, New York, USA eDepartment of Mathematics, University of Dortmund, 44221 Dortmund, Germany Received 3 May 1993; revised 10 January 1995
Abstract
Conditions are given for the existence of hamiltonian paths and cycles in the so-called Toeplitz graphs, i.e. simple graphs with a symmetric Toeplitz adjacency matrix.
Keywords: Toeplitz graph; Hamiltonian graph; Traceable graph
O. I n t r o d u c t i o n 2
An (n x n) matrix A = (au) is called a Toeplitz matrix if aij = ai+ 1,j+ t for each i,j = 1. . . . . n - 1. Toeplitz matrices are precisely those matrices that are constant along all diagonals parallel to the main diagonal, and thus a Toeplitz matrix is determined by its first row and column. Toeplitz matrices occur in a large variety of areas in pure and applied mathematics. For example, they often appear when differential or integral equations are discretized, they arise in physical data-processing applications, and in the theories of orthogonal polynomials, stationary processes, and moment problems; see Heinig and Rost [9]. Other references on Toeplitz matrices are Gohberg [8] and lohvidov [10]. *Corresponding author. 1Current address: Interdiszipliniires Zentrum fiir Wissenschaftliches Rechnen (IWR), Universitlit Heidelberg, Im Neuenheimer Feld 368, 69120 Heidelberg, Germany, E-mail: [email protected]. 2A section in Ch. 3 of the Ph.D. Thesis by Ren6 van Dal, entitled 'Special Cases of the Traveling Salesman Problem', Wolters-Noordhoff by, Groningen, The Netherlands, is based on this paper. 0012-365X/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSD10012-365X(95)O0111-5
R. van Dal et al./Discrete Mathematics 159 (1996) 69-81
70
T =
001001 000100 100010 010001 001000 100100
5
3
Fig. 1. A symmetric Toeplitz adjacency matrix T and the corresponding graph.
A T o e p l i t z g r a p h is a (undirected) graph with a symmetric Toeplitz adjacency matrix. Therefore, an (n x n) matrix B = (b~j) is the adjacency matrix of the Toeplitz graph G on n vertices if B is a 0 - 1 Toeplitz matrix, B is symmetric, and for all i , j = 1 . . . . , n the following holds: the edge { i , j } is in the edge set of G if and only if bij = bji = 1. In this paper we describe hamiltonian properties of Toeplitz graphs. The n distinct diagonals of an (n x n) symmetric Toeplitz adjacency matrix will be labeled 0, 1, 2, ..., n - 1. Diagonal 0 is the main diagonal and it contains only zeros, i.e. aii = 0 for all i = 1, ... ,n so that there are no loops in the Toeplitz graph. Let t l , t2 . . . . . tk be the diagonals containing ones (0 < tl < t2 < ... < tk < n). Then, the corresponding Toeplitz graph will be denoted by T , ( t l . . . . , t k ) . That is, T , ( t l . . . . . t k ) is the graph with vertex set 1,2 . . . . . n in which the edge { i , j } , 1 ~< i < j ~< n, occurs if and only if j - i = t l for some l, 1 ~< l ~< k. For example, let n = 6, k = 2, t~ = 2, and t2 = 5. Fig. 1 shows the symmetric Toeplitz adjacency matrix T and the Toeplitz graph T 6 (2, 5). Closely related to Toeplitz matrices are the so-called circulant matrices. An (n x n) matrix C is called a circulant m a t r i x if it is of the form f cO Cn-1
el Co
C2 C1
"" ".
Cn-2 Cn_ 3
Cn-1 ~ Cn-2[
\ cl
c2
c3
"'"
Cn 1
CO /
F o r each i , j = 1 . . . . , n and k = 0 , 1. . . . . n - 1 , all the elements ( i , j ) such that j - i = k (mod n) have the same value Ck; these elements form the so-called kth stripe of C. Obviously, a circulant matrix is determined by its first row (or column). It is clear that every circulant matrix is a Toeplitz matrix, but the converse is not necessarily true. Circulant matrices and their properties have been studied extensively in Davis
ES]. Several authors have formulated conditions for connectivity and Hamiltonicity of circulant (di)graphs, i.e. (di)graphs with a circulant adjacency matrix. Garfinkel [7] proved that the number of dicycles associated with the kth stripe of an (n x n) circulant matrix is given by gcd(k, n). Boesch and Tindell [1] characterized the circulant graphs which are connected and conjectured that all connected circulant graphs are
R. van Dal et aL /Discrete Mathematics 159 (1996) 6~81
71
hamiltonian. This conjecture has been proven by Burkard and Sandholzer [2]. Van D o o r n [6] derived an explicit expression for the connectivity of circulant digraphs. M e d o v a - D e m p s t e r [11-] considered the asymmetric T S P (Traveling Salesman Problem) for circulant matrices and conjectured that this problem is ,/V~-hard in the general case. Finally, Van der Veen et al. 1-13] described two heuristics for the TSP restricted to symmetric circulant matrices and showed that these two heuristics are superior to some well-known heuristics for solving the general symmetric TSP. Our motivation for considering hamiltonian properties of Toeplitz graphs is twofold. First, we wish to study these properties for graphs that have adjacency matrices from a broader class than that of the circulant adjacency matrices. Our second aim is to investigate which classes of subgraphs of the complete graph the TSP to be efficiently solvable (note that if the TSP for circulant matrices turns out to be ,Jf'.~-hard, then the T S P for Toeplitz matrices is also ~P'~-hard). At present there are only few examples of such classes of subgraphs. Cornu6jols et al. [-3, 4-] gave a polynomial-time algorithm for the T S P restricted to Halin graphs and Ratliff and Rosenthal [12] solved efficiently the TSP for a graph that models a rectangular warehouse. We are indebted to Gerard Sierksma for his moral support and for inviting Tudor Zamfirescu to visit the university of Groningen to participate in joint research. This co-operation resulted in the present paper.
1. Nonhamiltonian and nontraeeable Toeplitz graphs We start with a few simple results providing necessary conditions for a Toeplitz graph T, (tl . . . . . t , ) to be hamiltonian or traceable, where a graph is called traceable if it admits a hamiltonian path. Clearly, a necessary condition for traceability is connectedness and a necessary condition for Hamiltonicity is 2-connectedness. The following theorem gives a lower bound on the number of components of a Toeplitz graph.
Theorem 1. T . ( t l . . . . . tk) has at least gcd(t~ . . . . . tk) components. Proof. It will be shown that the vertices 1, ... ,gcd(tl, ... ,tk) are all in different components. Let u, v~ {1 . . . . . gcd(t~, ... ,tk)} and u :~ v. Assume that the vertices u and v are in the same component, i.e. there is a path joining u and v. So, there are 2i~ Z, i = 1. . . . . k such that
k U----V+ ~ )~iti. i=l
Therefore, there is a 2 ~ Z\{0} such that u = v + 2 gcd(tl . . . . . tk) which contradicts the assumption that u, v ~ {1 . . . . . gcd(tl . . . . . tk)} and u :~ v. [] Corollary 1. f f g c d { t l . . . . . tk) > 1, then T , ( t l . . . . . tk) is disconnected.
72
R. van Dal et al./Discrete Mathematics 159 (1996) 69-81
Note that T , ( t a . . . . . tk) can have more than gcd(q . . . . , t k ) components. For instance, consider T5 (3, 4) which has 2 components whereas gcd(3, 4) = 1. On the other hand, Burkard and Sandholzer [2] showed that if T , ( t l . . . . , tk) is a circulant graph, i.e. if h occurs then n - h also occurs for all l = 1. . . . , k, then the number of components is exactly gcd(tl, ..., tk). Theorem 2. I f there is a nonempty subset J of K = {1 . . . . ,k} such that (n-
tl) < gcd{tj l j ~ J}
(1)
i~K\J
then the Toeplitz graph T . ( t l . . . . . tk) is not 2-connected, and if there is a nonempty subset J of K = { 1. . . . . k } such that 2
(n-
ti) (k - 1)n i=1
7
1
6
2
5
3
4
Fig. 2. The Toeplitz graph Tv (3, 5, 6).
R. van Dal et a l . / D i s c r e t e Mathematics 159 (1996) 6 ~ 8 1
73
then T , ( t l . . . . . tk) is not 2-connected and if k
ti>(k-
1)n+ 1
i:l
then T, ( t l . . . . . tk ) is disconnected. Proof. The number of edges of T,(t~ . . . . . tk) is k
k
y ( n - t,)= k n i=1
ti. i=1
Therefore, under the first assumption the number of edges is less than n, and hence T , ( t l . . . . . tk) is not 2-connected, and under the second assumption the number of edges is less than n - 1, and hence T , ( t l , ... ,tk) is disconnected. [] Theorem 4. Consider the Toeplitz graph T , ( t l , t2) and let n >>-5. I f tl + t 2 < n < 3 t l
+t2
(3)
then T , ( t l , t2) is nonhamiltonian. I f tl >~ 3 and tl + t 2 + 2 < n < 3 t ~
+t2
(4)
then T , ( t l , t2) is nontraceable. Proof. (3) is equivalent to the existence of two vertices u, v such that 1 < ~ u < < , t a , n - t l 2, t2 > / n / 2 , tl o d d f o r all i = 2 . . . . , k. T , ( 2 , t 2. . . . , tk ) is h a m i l t o n i a n ! l a n d o n l y i f (n - ti + 1)/2, i = 2 . . . . . k, a r e n o t all o f t h e s a m e p a r i t y . P r o o f . (if) S u p p o s e t w o of the i n t e g e r s (n - tl + 1)/2, i = 2, . . . , k, say ~ a n d / 3 , are of different parity, for e x a m p l e e even a n d / 3 odd. T h e n (1, n - 2/3 + 2, n - 2/3 . . . . . 27 - 2, n - 1, n - 3 , 2 ~ - 4 , 2 ~ - 6 ..... 4,2, n-2e+3, n-2~+ 1, . . . , 2 / 3 1, n , n - 2 , 2/3 - 3, 2/3 - 5 . . . . ,3) is a h a m i l t o n i a n cycle of T , ( 2 , t2 . . . . . t k ) (see Fig. 9 for the case n = 2 0 , k - - 3 , t 2 = 11 a n d t 3 = 13). (only if) S u p p o s e t h a t T , (2, t2 . . . . . t k ) is h a m i l t o n i a n a n d all integers (n - ti + 1)/2, i = 2 . . . . . k, are even (the o d d case is a n a l o g o u s ) . L e t e = (n - t2 + 1)/2 a n d s h r i n k the s u b g r a p h of T , ( 2 , t2, ... , t k ) s p a n n e d b y the v e r t e x set {2~ + 1, 2c~ + 3, ... ,n - 1, 2, 4 . . . . . n - 2e} to a single vertex v. O b v i o u s l y , the new g r a p h H is also h a m i l t o n i a n . But H is b i p a r t i t e , since its v e r t e x set can be p a r t i t i o n e d i n t o the s u b s e t s {v, 1, 3 + t2, 5 , 7 + t2 . . . . ,n - t2 - 2, n} a n d {1 + t2, 3, 5 + re, 7 . . . . ,n - 2, n - t2} (see Fig. 10). M o r e o v e r , H has a n o d d n u m b e r of vertices, n a m e l y 2e + 1. H e n c e , H is n o n h a m i l t o n i a n , w h i c h l e a d s to a c o n t r a d i c t i o n . [] F o r t 2 o d d a n d s m a l l e n o u g h a T o e p l i t z g r a p h is a l w a y s h a m i l t o n i a n , as the f o l l o w i n g result shows.
~
10 12 14 5i 7-
/
2t'-.. t 13
\
4 ~.~,. "~ 15 6 ~ . . " ~ 17 ~ 19
\ ~
- 16 -18
Fig. 9. The Toeplitz graph T2o(2 , 11, 13).
78
R. van Dal et al./Discrete Mathematics 159 (1996) 6941
vii Fig. 10. Graph H.
1
t2-1-2
2
2t2-1
t2+l
n-t 2
n-2t 2 +2
rL-1
¢~-t2-1
n
Fig. 11. T,(2,4m + 1) with n even.
1
t2+2
2
n-2t2+l
t2+l
2t 2
n-t 2
n-t2-1
n-1
n
Fig. 12. T,(2,4m - 1) with n even.
T h e o r e m 8. Let t2 = e (mod 4), where e = + 1. I f n is even and t2 ~ 10, and, by T h e o r e m 8, T, (2, t2) is hamiltonian. The 3 shows that the b o u n d (n + 3)/4 mentioned in the
4. Other classes of Toeplitz graphs In the case that k = 2 and tl, t2 satisfy the inequality t 1 4- 2t2 ~> n, we are able to characterize the Toeplitz graphs that are hamiltonian. Since the situation for t~ -- 1 has been completely described in T h e o r e m 5, we now deal with t~ ~> 2. In addition, we shall assume gcd(tl, t2) ~ 1, for otherwise T , ( t l , t 2 ) is disconnected, by Corollary 1.
Theorem 9. L e t k = 2, tl /> 2, gcd(tl, t2) = 1, and suppose tl 4- 2t2 >1 n. T , ( h ,
t2)
is
h a m i h o n i a n if and only if (n - t2)/tl is an odd integer.
Proof. (only if) F o r v = 1. . . . ,t~ let q = q(v) be the largest integer such that v + qtl + t2 2. Consider two subgraphs of T , ( t l , t2), namely G I = ( V 1 , E 1 ) with Va={1,2 .... ,tl+t2} and E ~ = { { i , j } E E [ i , j ~ V I } , and G z = ( V 2 , E 2 ) with 11"2-- {t~ + t2 + 1, ... ,n} and E2 = {{i,j} ~ E [ i , j ~ V}. By Theorem 9 and by the induction hypothesis, both Ga and G2 are hamiltonian. Moreover, G~ is itself a cycle and h + t2 + 1 has degree 2 in G2. To obtain a hamiltonian cycle in T , ( t l , t 2 ) we remove two edges, namely {1 + tl, 1 + 2tl} from the cycle Gt and {1 + tl -+- t2, 1 + 2t~ + t2} from the
R. van Dal et al./Discrete Mathematics 159 (1996) 69-81
81
. H
Fig. 16. Patching of the hamiltonian cycles in G, and GB.
h a m i l t o n i a n cycle in G2, a n d add the two edges {1 + t t , 1 + t l + t2} a n d {1 + 2tl, 1 + 2tl + tz} from E \ ( E 1 u E 2 ) to c o n n e c t the two resulting h a m i l t o n i a n paths in G1 a n d G 2.
[]
Acknowledgements T. Zamfirescu thankfully acknowledges generous s u p p o r t from the University of G r o n i n g e n in 1990 a n d from the E u r o p e a n C o m m u n i t y d u r i n g the C O S T m o b i l i t y action C I P A - C T - 9 3 - 1 5 4 7 .
References [-l] F. Boesch and R. Tindell, Circulants and their connectivities, ]. Graph Theory 8 (1984) 487-499. [2] R.E. Burkard and W. Sandholzer, Efficientlysolvable special cases of bottleneck travelling salesman problems, Discrete Appl. Math. 32 (1991) 61-67. [3] G. Cornu6jols, D. Naddef and W.R. Pulleyblank, Halin graphs and the travelling salesman problem, Math. Programming 26 (1983) 287-294. [4] G. Cornu+jols, D. Naddef and W. Pulleyblank, "['hetraveling salesman problem in graphs with 3-edge cutsets, J. ACM 32 (1985) 383 410. [5] Ph.J. Davis, Circulant Matrices (Wiley, New York, 1979). [6] E.A. van Doorn, Connectivity of circulant digraphs, J, Graph Theory 10 (1986) 9-14. [7] R.S.Garfinkel, Minimizing wallpaper waste, part l:aclassoftravelingsalesmanproblems,Oper. Res. 25 (1977) 741 751. [8] I. Gohberg (ed.), Toeplitz Centennial (Birkh~iuser,Boston, 1982). [9] G. Heinig and K. Rost, Algebraic methods for Toeplitz-like matrices and operators (Birkh~iuser, Boston, 1984). [10] I.S. Iohvidov, Hankel and Toeplitz matrices and forms algebraic theory (Birkh~iuser,Boston, 1982). [11] E.A. Medova-Dempster, The circulant traveling salesman problem, Technical Report, Department of Mathematics, University of Pisa, 1988. [12] H.D. Ratliff and A.S. Rosenthal, Order-picking in a rectangular warehouse: a solvable case of the traveling salesman problem, Oper. Res. 31 (1983) 507-521. [13] J.A.A. van der Veen, R. van Dal and G. Sierksma, The symmetric circulant traveling salesman problem, Research Memorandum No. 429, Institute of Economic Research, University of Groningen, 1991.
Discrete Mathematics 159 (1996) 69-81
Hamiltonian properties of Toeplitz graphs Ren6 van D a P '1, G e r t Tijssen a, Zsolt T u z a b, J a c k A.A. v a n der Veen c, C h r i s t i n a Z a m f i r e s c u d, T u d o r Z a m f i r e s c u e'* aDepartment of Econometrics, University of Groningen, P.O. Box 800, 9700 A V Groningen, Netherlands bComputer and Automation Institute, Hungarian Academy of Sciences, H-1111 Budapest, Kende u. 13-17, Hungary CDepartment of Quantitative Methods, Nijenrode, Netherlands School of Business, Straatweg 25, 3621 BG Breukelen, Netherlands aDepartment of Computer Science, Hunter College, City University of New York, New York, USA eDepartment of Mathematics, University of Dortmund, 44221 Dortmund, Germany Received 3 May 1993; revised 10 January 1995
Abstract
Conditions are given for the existence of hamiltonian paths and cycles in the so-called Toeplitz graphs, i.e. simple graphs with a symmetric Toeplitz adjacency matrix.
Keywords: Toeplitz graph; Hamiltonian graph; Traceable graph
O. I n t r o d u c t i o n 2
An (n x n) matrix A = (au) is called a Toeplitz matrix if aij = ai+ 1,j+ t for each i,j = 1. . . . . n - 1. Toeplitz matrices are precisely those matrices that are constant along all diagonals parallel to the main diagonal, and thus a Toeplitz matrix is determined by its first row and column. Toeplitz matrices occur in a large variety of areas in pure and applied mathematics. For example, they often appear when differential or integral equations are discretized, they arise in physical data-processing applications, and in the theories of orthogonal polynomials, stationary processes, and moment problems; see Heinig and Rost [9]. Other references on Toeplitz matrices are Gohberg [8] and lohvidov [10]. *Corresponding author. 1Current address: Interdiszipliniires Zentrum fiir Wissenschaftliches Rechnen (IWR), Universitlit Heidelberg, Im Neuenheimer Feld 368, 69120 Heidelberg, Germany, E-mail: [email protected]. 2A section in Ch. 3 of the Ph.D. Thesis by Ren6 van Dal, entitled 'Special Cases of the Traveling Salesman Problem', Wolters-Noordhoff by, Groningen, The Netherlands, is based on this paper. 0012-365X/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSD10012-365X(95)O0111-5
R. van Dal et al./Discrete Mathematics 159 (1996) 69-81
70
T =
001001 000100 100010 010001 001000 100100
5
3
Fig. 1. A symmetric Toeplitz adjacency matrix T and the corresponding graph.
A T o e p l i t z g r a p h is a (undirected) graph with a symmetric Toeplitz adjacency matrix. Therefore, an (n x n) matrix B = (b~j) is the adjacency matrix of the Toeplitz graph G on n vertices if B is a 0 - 1 Toeplitz matrix, B is symmetric, and for all i , j = 1 . . . . , n the following holds: the edge { i , j } is in the edge set of G if and only if bij = bji = 1. In this paper we describe hamiltonian properties of Toeplitz graphs. The n distinct diagonals of an (n x n) symmetric Toeplitz adjacency matrix will be labeled 0, 1, 2, ..., n - 1. Diagonal 0 is the main diagonal and it contains only zeros, i.e. aii = 0 for all i = 1, ... ,n so that there are no loops in the Toeplitz graph. Let t l , t2 . . . . . tk be the diagonals containing ones (0 < tl < t2 < ... < tk < n). Then, the corresponding Toeplitz graph will be denoted by T , ( t l . . . . , t k ) . That is, T , ( t l . . . . . t k ) is the graph with vertex set 1,2 . . . . . n in which the edge { i , j } , 1 ~< i < j ~< n, occurs if and only if j - i = t l for some l, 1 ~< l ~< k. For example, let n = 6, k = 2, t~ = 2, and t2 = 5. Fig. 1 shows the symmetric Toeplitz adjacency matrix T and the Toeplitz graph T 6 (2, 5). Closely related to Toeplitz matrices are the so-called circulant matrices. An (n x n) matrix C is called a circulant m a t r i x if it is of the form f cO Cn-1
el Co
C2 C1
"" ".
Cn-2 Cn_ 3
Cn-1 ~ Cn-2[
\ cl
c2
c3
"'"
Cn 1
CO /
F o r each i , j = 1 . . . . , n and k = 0 , 1. . . . . n - 1 , all the elements ( i , j ) such that j - i = k (mod n) have the same value Ck; these elements form the so-called kth stripe of C. Obviously, a circulant matrix is determined by its first row (or column). It is clear that every circulant matrix is a Toeplitz matrix, but the converse is not necessarily true. Circulant matrices and their properties have been studied extensively in Davis
ES]. Several authors have formulated conditions for connectivity and Hamiltonicity of circulant (di)graphs, i.e. (di)graphs with a circulant adjacency matrix. Garfinkel [7] proved that the number of dicycles associated with the kth stripe of an (n x n) circulant matrix is given by gcd(k, n). Boesch and Tindell [1] characterized the circulant graphs which are connected and conjectured that all connected circulant graphs are
R. van Dal et aL /Discrete Mathematics 159 (1996) 6~81
71
hamiltonian. This conjecture has been proven by Burkard and Sandholzer [2]. Van D o o r n [6] derived an explicit expression for the connectivity of circulant digraphs. M e d o v a - D e m p s t e r [11-] considered the asymmetric T S P (Traveling Salesman Problem) for circulant matrices and conjectured that this problem is ,/V~-hard in the general case. Finally, Van der Veen et al. 1-13] described two heuristics for the TSP restricted to symmetric circulant matrices and showed that these two heuristics are superior to some well-known heuristics for solving the general symmetric TSP. Our motivation for considering hamiltonian properties of Toeplitz graphs is twofold. First, we wish to study these properties for graphs that have adjacency matrices from a broader class than that of the circulant adjacency matrices. Our second aim is to investigate which classes of subgraphs of the complete graph the TSP to be efficiently solvable (note that if the TSP for circulant matrices turns out to be ,Jf'.~-hard, then the T S P for Toeplitz matrices is also ~P'~-hard). At present there are only few examples of such classes of subgraphs. Cornu6jols et al. [-3, 4-] gave a polynomial-time algorithm for the T S P restricted to Halin graphs and Ratliff and Rosenthal [12] solved efficiently the TSP for a graph that models a rectangular warehouse. We are indebted to Gerard Sierksma for his moral support and for inviting Tudor Zamfirescu to visit the university of Groningen to participate in joint research. This co-operation resulted in the present paper.
1. Nonhamiltonian and nontraeeable Toeplitz graphs We start with a few simple results providing necessary conditions for a Toeplitz graph T, (tl . . . . . t , ) to be hamiltonian or traceable, where a graph is called traceable if it admits a hamiltonian path. Clearly, a necessary condition for traceability is connectedness and a necessary condition for Hamiltonicity is 2-connectedness. The following theorem gives a lower bound on the number of components of a Toeplitz graph.
Theorem 1. T . ( t l . . . . . tk) has at least gcd(t~ . . . . . tk) components. Proof. It will be shown that the vertices 1, ... ,gcd(tl, ... ,tk) are all in different components. Let u, v~ {1 . . . . . gcd(t~, ... ,tk)} and u :~ v. Assume that the vertices u and v are in the same component, i.e. there is a path joining u and v. So, there are 2i~ Z, i = 1. . . . . k such that
k U----V+ ~ )~iti. i=l
Therefore, there is a 2 ~ Z\{0} such that u = v + 2 gcd(tl . . . . . tk) which contradicts the assumption that u, v ~ {1 . . . . . gcd(tl . . . . . tk)} and u :~ v. [] Corollary 1. f f g c d { t l . . . . . tk) > 1, then T , ( t l . . . . . tk) is disconnected.
72
R. van Dal et al./Discrete Mathematics 159 (1996) 69-81
Note that T , ( t a . . . . . tk) can have more than gcd(q . . . . , t k ) components. For instance, consider T5 (3, 4) which has 2 components whereas gcd(3, 4) = 1. On the other hand, Burkard and Sandholzer [2] showed that if T , ( t l . . . . , tk) is a circulant graph, i.e. if h occurs then n - h also occurs for all l = 1. . . . , k, then the number of components is exactly gcd(tl, ..., tk). Theorem 2. I f there is a nonempty subset J of K = {1 . . . . ,k} such that (n-
tl) < gcd{tj l j ~ J}
(1)
i~K\J
then the Toeplitz graph T . ( t l . . . . . tk) is not 2-connected, and if there is a nonempty subset J of K = { 1. . . . . k } such that 2
(n-
ti) (k - 1)n i=1
7
1
6
2
5
3
4
Fig. 2. The Toeplitz graph Tv (3, 5, 6).
R. van Dal et a l . / D i s c r e t e Mathematics 159 (1996) 6 ~ 8 1
73
then T , ( t l . . . . . tk) is not 2-connected and if k
ti>(k-
1)n+ 1
i:l
then T, ( t l . . . . . tk ) is disconnected. Proof. The number of edges of T,(t~ . . . . . tk) is k
k
y ( n - t,)= k n i=1
ti. i=1
Therefore, under the first assumption the number of edges is less than n, and hence T , ( t l . . . . . tk) is not 2-connected, and under the second assumption the number of edges is less than n - 1, and hence T , ( t l , ... ,tk) is disconnected. [] Theorem 4. Consider the Toeplitz graph T , ( t l , t2) and let n >>-5. I f tl + t 2 < n < 3 t l
+t2
(3)
then T , ( t l , t2) is nonhamiltonian. I f tl >~ 3 and tl + t 2 + 2 < n < 3 t ~
+t2
(4)
then T , ( t l , t2) is nontraceable. Proof. (3) is equivalent to the existence of two vertices u, v such that 1 < ~ u < < , t a , n - t l 2, t2 > / n / 2 , tl o d d f o r all i = 2 . . . . , k. T , ( 2 , t 2. . . . , tk ) is h a m i l t o n i a n ! l a n d o n l y i f (n - ti + 1)/2, i = 2 . . . . . k, a r e n o t all o f t h e s a m e p a r i t y . P r o o f . (if) S u p p o s e t w o of the i n t e g e r s (n - tl + 1)/2, i = 2, . . . , k, say ~ a n d / 3 , are of different parity, for e x a m p l e e even a n d / 3 odd. T h e n (1, n - 2/3 + 2, n - 2/3 . . . . . 27 - 2, n - 1, n - 3 , 2 ~ - 4 , 2 ~ - 6 ..... 4,2, n-2e+3, n-2~+ 1, . . . , 2 / 3 1, n , n - 2 , 2/3 - 3, 2/3 - 5 . . . . ,3) is a h a m i l t o n i a n cycle of T , ( 2 , t2 . . . . . t k ) (see Fig. 9 for the case n = 2 0 , k - - 3 , t 2 = 11 a n d t 3 = 13). (only if) S u p p o s e t h a t T , (2, t2 . . . . . t k ) is h a m i l t o n i a n a n d all integers (n - ti + 1)/2, i = 2 . . . . . k, are even (the o d d case is a n a l o g o u s ) . L e t e = (n - t2 + 1)/2 a n d s h r i n k the s u b g r a p h of T , ( 2 , t2, ... , t k ) s p a n n e d b y the v e r t e x set {2~ + 1, 2c~ + 3, ... ,n - 1, 2, 4 . . . . . n - 2e} to a single vertex v. O b v i o u s l y , the new g r a p h H is also h a m i l t o n i a n . But H is b i p a r t i t e , since its v e r t e x set can be p a r t i t i o n e d i n t o the s u b s e t s {v, 1, 3 + t2, 5 , 7 + t2 . . . . ,n - t2 - 2, n} a n d {1 + t2, 3, 5 + re, 7 . . . . ,n - 2, n - t2} (see Fig. 10). M o r e o v e r , H has a n o d d n u m b e r of vertices, n a m e l y 2e + 1. H e n c e , H is n o n h a m i l t o n i a n , w h i c h l e a d s to a c o n t r a d i c t i o n . [] F o r t 2 o d d a n d s m a l l e n o u g h a T o e p l i t z g r a p h is a l w a y s h a m i l t o n i a n , as the f o l l o w i n g result shows.
~
10 12 14 5i 7-
/
2t'-.. t 13
\
4 ~.~,. "~ 15 6 ~ . . " ~ 17 ~ 19
\ ~
- 16 -18
Fig. 9. The Toeplitz graph T2o(2 , 11, 13).
78
R. van Dal et al./Discrete Mathematics 159 (1996) 6941
vii Fig. 10. Graph H.
1
t2-1-2
2
2t2-1
t2+l
n-t 2
n-2t 2 +2
rL-1
¢~-t2-1
n
Fig. 11. T,(2,4m + 1) with n even.
1
t2+2
2
n-2t2+l
t2+l
2t 2
n-t 2
n-t2-1
n-1
n
Fig. 12. T,(2,4m - 1) with n even.
T h e o r e m 8. Let t2 = e (mod 4), where e = + 1. I f n is even and t2 ~ 10, and, by T h e o r e m 8, T, (2, t2) is hamiltonian. The 3 shows that the b o u n d (n + 3)/4 mentioned in the
4. Other classes of Toeplitz graphs In the case that k = 2 and tl, t2 satisfy the inequality t 1 4- 2t2 ~> n, we are able to characterize the Toeplitz graphs that are hamiltonian. Since the situation for t~ -- 1 has been completely described in T h e o r e m 5, we now deal with t~ ~> 2. In addition, we shall assume gcd(tl, t2) ~ 1, for otherwise T , ( t l , t 2 ) is disconnected, by Corollary 1.
Theorem 9. L e t k = 2, tl /> 2, gcd(tl, t2) = 1, and suppose tl 4- 2t2 >1 n. T , ( h ,
t2)
is
h a m i h o n i a n if and only if (n - t2)/tl is an odd integer.
Proof. (only if) F o r v = 1. . . . ,t~ let q = q(v) be the largest integer such that v + qtl + t2 2. Consider two subgraphs of T , ( t l , t2), namely G I = ( V 1 , E 1 ) with Va={1,2 .... ,tl+t2} and E ~ = { { i , j } E E [ i , j ~ V I } , and G z = ( V 2 , E 2 ) with 11"2-- {t~ + t2 + 1, ... ,n} and E2 = {{i,j} ~ E [ i , j ~ V}. By Theorem 9 and by the induction hypothesis, both Ga and G2 are hamiltonian. Moreover, G~ is itself a cycle and h + t2 + 1 has degree 2 in G2. To obtain a hamiltonian cycle in T , ( t l , t 2 ) we remove two edges, namely {1 + tl, 1 + 2tl} from the cycle Gt and {1 + tl -+- t2, 1 + 2t~ + t2} from the
R. van Dal et al./Discrete Mathematics 159 (1996) 69-81
81
. H
Fig. 16. Patching of the hamiltonian cycles in G, and GB.
h a m i l t o n i a n cycle in G2, a n d add the two edges {1 + t t , 1 + t l + t2} a n d {1 + 2tl, 1 + 2tl + tz} from E \ ( E 1 u E 2 ) to c o n n e c t the two resulting h a m i l t o n i a n paths in G1 a n d G 2.
[]
Acknowledgements T. Zamfirescu thankfully acknowledges generous s u p p o r t from the University of G r o n i n g e n in 1990 a n d from the E u r o p e a n C o m m u n i t y d u r i n g the C O S T m o b i l i t y action C I P A - C T - 9 3 - 1 5 4 7 .
References [-l] F. Boesch and R. Tindell, Circulants and their connectivities, ]. Graph Theory 8 (1984) 487-499. [2] R.E. Burkard and W. Sandholzer, Efficientlysolvable special cases of bottleneck travelling salesman problems, Discrete Appl. Math. 32 (1991) 61-67. [3] G. Cornu6jols, D. Naddef and W.R. Pulleyblank, Halin graphs and the travelling salesman problem, Math. Programming 26 (1983) 287-294. [4] G. Cornu+jols, D. Naddef and W. Pulleyblank, "['hetraveling salesman problem in graphs with 3-edge cutsets, J. ACM 32 (1985) 383 410. [5] Ph.J. Davis, Circulant Matrices (Wiley, New York, 1979). [6] E.A. van Doorn, Connectivity of circulant digraphs, J, Graph Theory 10 (1986) 9-14. [7] R.S.Garfinkel, Minimizing wallpaper waste, part l:aclassoftravelingsalesmanproblems,Oper. Res. 25 (1977) 741 751. [8] I. Gohberg (ed.), Toeplitz Centennial (Birkh~iuser,Boston, 1982). [9] G. Heinig and K. Rost, Algebraic methods for Toeplitz-like matrices and operators (Birkh~iuser, Boston, 1984). [10] I.S. Iohvidov, Hankel and Toeplitz matrices and forms algebraic theory (Birkh~iuser,Boston, 1982). [11] E.A. Medova-Dempster, The circulant traveling salesman problem, Technical Report, Department of Mathematics, University of Pisa, 1988. [12] H.D. Ratliff and A.S. Rosenthal, Order-picking in a rectangular warehouse: a solvable case of the traveling salesman problem, Oper. Res. 31 (1983) 507-521. [13] J.A.A. van der Veen, R. van Dal and G. Sierksma, The symmetric circulant traveling salesman problem, Research Memorandum No. 429, Institute of Economic Research, University of Groningen, 1991.
