A lower bound for the optimal crossing-free Hamiltonian cycle problem

Discrete & Computational Geometry, Dec 1987

Consider a drawing in the plane ofK n , the complete graph onn vertices. If all edges are restricted to be straight line segments, the drawing is called rectilinear. Consider a Hamiltonian cycle in a drawing ofK n . If no pair of the edges of the cycle cross, it is called a crossing-free Hamiltonian cycle (cfhc). Let Φ(n) represent the maximum number of cfhc's of any drawing ofK n , and\(\bar \Phi\)(n) the maximum number of cfhc's of any rectilinear drawing ofK n . The problem of determining Φ(n) and\(\bar \Phi\)(n), and determining which drawings have this many cfhc's, is known as the optimal cfhc problem. We present a brief survey of recent work on this problem, and then, employing a recursive counting argument based on computer enumeration, we establish a substantially improved lower bound for Φ(n) and\(\bar \Phi\)(n). In particular, it is shown that\(\bar \Phi\)(n) is at leastk × 3.2684 n . We conjecture that both Φ(n) and\(\bar \Phi\)(n) are at mostc × 4.5 n .

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A lower bound for the optimal crossing-free Hamiltonian cycle problem

Discrete Comput Geom Ryan B. Hayward 0 0 School of Computer Science, McGill University , 805 SherbrookeSt. West, Montreal, Quebec , Canada H3A2K6 Consider a drawing in the plane of K~, the complete graph on n vertices. If all edges are restricted to be straight line segments, the drawing is called rectilinear. Consider a Hamiltonian cycle in a drawing of K,. If no pair of the edges of the cycle cross, it is called a crossing-free Hamiltonian cycle (cfhc). Let ~(n) represent the maximum number of cfhc's of any drawing of K,, and ~(n) the maximum number of cfhc's of any rectilinear drawing of K,. The problem of determining q~(n) and ~(n), and determining which drawings have this many cfhc's, is known as the optimal ofhe problem. We present a brief survey of recent work on this problem, and then, employing a recursive counting argument based on computer enumeration, we establish a substantially improved lower bound for q~(n) and ~(n), In particular, it is shown that ~(n) is at least k x 3.2684". We conjecture that both • (n) and t~(n) are at most cx4.5 ~. - *This research, part of which was conducted at Queen's University, was supported by an N.S.E.R.C. postgraduatescholarship. o f Kn have ~ ( n ) (respectively ~ ( n ) ) cfhc's. Such drawings will be referred to as cfhc-optimal (respectively rectilinear cfhc-optimat) drawings. Figure 1 shows a drawing of Ks which is both cfhc-optimal and rectilinear cfhc-optimal. The drawing has 29 cfhc's; the other 91 Hamiltonian cycles all have at least one crossing. Let ~,(n) (respectively ~(n)) refer to the minimum number of crossings of any drawing (respectively rectilinear drawing) of Kn. The optimal crossing problem, also known as the crossing number problem, is to determine the values of u(n) and ~(n), and to find which drawings attain this number of crossings. The optimal crossing problem is related to the optimal cfhc problem in that, in general, drawings with fewer crossings have more cfhc's. However, this is not always the case (e.g., see [H]). Although the optimal crossing problem has been extensively studied (see [EG] or [G]), exact values for u(n) and ~(n) are not known for n > 1 0 . The optimal cfhc problem was first explored by Newborn and Moser [NM]. They were able to determine qb(n) and ~ ( n ) exactly for n from 3 to 6, and established lower bounds for other small values o f n. Later we extended this list of lower bounds [HI. The following is a list of the current best lower bounds for d~(n) and ~ ( n ) , for n up to 13 (values for n up to 8 were established in [NM], all others are taken from [H]): Be~ known lower bounds n • (n) ~(n) 3 1 ! 4 3 3 5 8 8 6 29 29 7 92 96 8 339 399 9 1252 1461 10 4956 6354 11 18383 24687 12 75 231 110 162 13 306446 446798 The first bounds for q~(n) or ~ ( n ) for arbitrary n were established by Newborn and Moser, who showed that ~ x 10n/3<_ ~(n)_< 2 x 6~-2 x [ ~ J !, where 101/3- 2.1544. The u p p e r b o u n d was substantially improved by Ajtai et al. [ACNS], who showed that every planar drawing o f any graph with n vertices contains at most 10 000 000 000 000" crossing-free subgraphs. Thus both ~ ( n ) and ~ ( n ) are exponential in n. The lower b o u n d was first improved by Akl [A], who showed that d~ < ~ ( n ) , where d, is asymptotically k x ( 5 + 3 Xv~) n/3, with k a constant and ( 5 + 3 x ~/~)I/3 -- 2.2707. In this paper, generalizing Akl's approach, we show how the lower b o u n d can be substantially improved by counting a subset of the cfhc's of a certain drawing TS~ of Kn. We prove that fn < ~ ( n ) , where f~ is asymptotically k x 3.2684 ~. 2. An Improved Lower Bound for ¢P(n) In this section we describe a certain rectilinear drawing TS. o f K., and then count a subset o f its cfhc's. This gives a new lower bound for ~ ( n ) . 2.1. A Description o f the Drawing TSn The " T S " in TSn is mnemonic for "trilateral spiral". Roughly speaking, the vertices of TS, can be thought of as resting on three gently spiralled arcs emanating from the origin. More precisely, let arc A be the arc o f the circle centered at the point in the plane with Cartesian coordinates (x, 2) and joining (in clockwise order) the points (0, 1) a n d (0, 3), where x > - 7 / 4 ~ . Arcs B a n d C are formed by rotating arc A respectively 120 and 240 ° clockwise a b o u t the origin, namely the point (0, 0). Place vertices 1,4, 7 , . . . on arc A, vertices 2, 5, 8 , . . . on arc B, and vertices 3, 6, 9 , . . . o n arc C, so that if v a n d w are on the same arc, and v < w, then v is closer to the origin than w (see Fig. 2). Figures 1, 3, and 4 show drawings o f TS6, TS9, a n d TS~2, respectively. The reason for c h o o s i n g x as described a b o v e is to ensure that the line segment joining the far end o f arc A to the near end o f arc C does not intersect arc A in a n y other point. In fact, the arcs are constructed so that any line segment joining points on two different arcs intersects each o f the two arcs in exactly one point, a n d does n o t intersect the third arc. Let a = ["~7 ' b = r"+'l L 3 J' c = [7] ' a n d relabel vertices 1, 4 . . . . , 3 a - 2 as A~, A 2 , . • . , Aa, vertices 2, 5. . . . , 3 b - 1 as B I , B 2 , . . . , Bb, and vertices 3, 6 , . . . , 3c as C~, C2 . . . . , Co. Then the following (3,0) arc A (1,o) is a description of all crossings of TSn: The number of crossings is a + b + c + a b b c c a + a b c a, w h e r e ( y ) is defined as 0 if x < y . Thus, the total number of crossings of TSn is for n congruent to 0, 1, 2 (mod 3), respectively. 2.2. Counting cfhc's of TS. Let cfhc (TS.) represent the number of cfhc's of TS.. We are unable to determine cfhc (TS.) explicitly for arbitrary n. However, by counting a proper subset of the cfhc's of T S . , we have established a lower bound for cfhc (TS.), which gives an improved lower bound for ~ ( n ) . Our counting argument is inductive, and relies on the fact that in any drawing of T S . , any consecutive set of r vertices induces a drawing isomorphic to TSr (two drawings o f Kn are isomorphic if the vertices of one can be relabeled so that both drawings have the same set of crossings). Thus it follows that in a drawing o f TS.+k, the drawing induced by vertices 1 to n is isomorphic to TS.. We will count cfhc's o f TS.+k by counting cfhc's of T S . , and then enumerating various ways in which cfhc's of TS. give rise to cfhc's o f TS.+k. We classify each cfhc o f TS. according to which o f the three outermost (convex hull) edges and which o f the three innermost edges the cfhc contains. In T S . , X, Y, and Z will represent, respectively, the edges (n - 2, n - 1), (n - 1, n), and (n - 2 , n) and x, y, and z will represent the edges (1, 2), (2, 3), and (1, 3). We will use y to represent cfhc's. Thus a y(X, n) will represent a cfhc of TS. that includes the edge ( n - 2 , n - 1 ) but neither edge ( n - l , n ) nor ( n - 2 , n), A y(yz, n) will represent a cfhc of TS. that includes the edges (2, 3) and (1, 3) but not edge (1, 2). We will ignore cfhc's which contain all or none of either the outermost or innermost edges. We create cfhc's of TS,+k by starting with a cfhc o f TS, on vertices 1 to n, removing either one or two of its outermost edges, and then joining the resulting crossing-free path to vertices n + 1 to n + k. For k = 1 and 2 we enumerate by hand all the possible ways in which this can be done. For k - 3 we show how this can be done in a more systematic way (and in a way which allows for computer enumeration). 2.2.1. Case k = I: creating cfhc's of TS,+, from cfhc's of TS,. Figure 5 shows all nine ways in which cfhc's o f TSn give rise to cfhc's o f TSn+I upon removal o f an outermost edge. (Only vertices n - 2 to n + 1 of TSn+t are shown in Fig. 5. The dashed line in the figures represents that part of the cfhc which visits vertices 1 to n - 3 . ) In particular, each y(X, n) gives rise to a y(Z, n + 1), each y(Y, n) gives rise to a y(YZ, n + l ) , each 7(Z, n) gives rise to a y(Y, n + l ) , each y(YZ, n) gives rise to a y(YZ, n + l ) and a y(XY, n + l ) , each y(ZX, n) gives rise to a y ( IF, n + 1) and a y(Z, n + 1), each y(XY, n) gives rise to a y(YZ, n + l ) and a y(ZX, n + l ) . For 1) = X, Y, Z, YZ, ZX, XY, let cfhc (l-l, n) represent the number of y(fl, n), and let t~ be the six element vector whose components are cfhc (fl, n). Then we have shown that t.+~>- N~ x t., (1) ~_...~ ~_,... ~(Y'~+~) -Y(Y,-) ,..~-" n 4"-" n--t ,],, " -~(YZ,.) .~(vz,.) ~ . / , -r (zx,.) A ,~(zx.n) l ~ k -~(xY,,) A -~(z,n+l) ,~(YZ,n+l) 3' (YZ,n+I) L ~ (XY,n+l) / ~ (Y,n+l) "~(Z,n+l) "r (ZX,n+l) "7(YZ,n+I) where ALowerBoundfortheOptimalcfhcProblem From (1) it followsthat cfhc(TS,) is asymptoticallyat least c x r~',where c is someconstant and r, is the dominanteigenvalueof N,, namely(to fourdecimal places) 1.8124. 2.2.2. Case k=2: creating cfhc's of TS.+2 from cfhc's of TS.. Figure 6 shows all ways in which cfhc's of TS.+2 are created from cfhc's of TS.. (Only vertices 3"(YZ,n) / , 3"(x..) "7(Y,n) 3"(z,n) "7(YZ,n) "7(YZ,n) 3"(ZX,n) -~(ZX,n) 3"(zx,n) 3"(xY..) "t(xY, n) 3"(XT,n) where (2.1) N~= t.+2-->-N2 x t., 0 1 and where /6 is the 6 by 6 identity matrix. The value of r2 is (to four decimal places) 2.1215. 2.2.3, Case k >-3: creating cfhc"s of TS,+k from cfhc's of TS,. We now show how to enumerate the ways in which cfhc's o f TSn+k can be created from cfhc's o f T S , , without having to draw figures corresponding to those shown in Figs. 5 and 6, A cfhc o f TSn÷k is created by taking a drawing of TS,+k, placing a cfhc o f T S , on vertices 1 to n, removing one or two o f its outermost edges, placing a cfhc of TS3÷k on vertices n - 2 to n + k, and then removing one or two o f the latter cfhc's innermost edges, so that each of the edges ( n - 2 , n - 1), ( n - 1, n), and ( n - 2 ) will have been removed from either the former or latter cfhc. For example, Fig. 7 shows how a cfhc of T S . is created by drawing a y(X, 8) on vertices 1 to 8, removing the edge X = (6.7), drawing a y(yz, 6) on vertices 6 to 11, and removing the edges y = (7, 8) and z = (6, 8). The following is a summary of all ways in which cfhc's of TS.+k can be thus created: Cfhc on vertices l t o n 3"(X, n) y( Y, n) 3,(Z, n) 3"(YZ, n) y( YZ, n) 3"(YZ, n) y(ZX, n) y(zx, n) v(ZX, n) y(XY, n) y(XY, n) y(XY, n) Edge(s) removed from cfhc on l . . . n Let T. be the 6 by 6 matrix whose entries are T.(fl, a), and let Q= .,o, 9 "tl* "r(x,s) 7 (y~,S) 7 (X,S) m i n u s edge X "7(yz,6) minus edges y and z Then the preceding inequality can be written in matrix form, namely t,+k>--NkXt,, where N k = T3+kXQ. (k.1) As before, we can improve slightly on this inequality by creating cfhc's of TS,÷k from cfhc's o f T S , , T S , + ~ , . . . , TS,+k_~ (i.e., not just from TSn). This yields the following: tn+k> M I x t . + k _ l + M2 × t n + k _ 2 + • • ' + M k × t,, (k.2) where Mk = Nk - ( N , x M,_, + N2 x Mk-2+ • • • + N k - , x M , ) . The matrices T3+kwere determined by computer enumeration, for k = 3 to 11. A LowerBound for the Optimal cfhc Problem Programs were written in C and run on a Vax 11-750 with operating system Unix 4.2. Time constraints prevented further computations. For successive values o f k, the amount of c.p.u, time used increased by a factor of about 6. As approximately 50 hours of c.p.u, time were required for k = 11, about 300 hours might be necessary for the case k = 12. As was the case with (2.2), the inequality (k.2) can be written as an inequality involving the single 6k by 6k matrix Pk, where P k = M1 I6 0 M2 "'" 0 0 . . . " ' " Mk-1 0 Mk 1 0 | 16 0 J The dominant eigenvalue rE of Pk gives the asymptotic rate of growth of the lower bound for cfhc (TS,) as determined by (k.2). The matrices T3+k and the eigenvdlues rk are all given in the Appendix. The best lower bound (to four decimal places), achieved with k = 11, is cfhc (TS,) --- c x 3.2684 ~. Thus it follows immediately that ~ ( n ) >- c x 3.2684". 3. Open Problems We have established an improved lower bound for ~ ( n ) , namely by counting only a proper subset o f the cfhc's o f TS,. Thus determining cfhc (TS,) explicitly or even asymptotically is still open. Extrapolating the values rk (see the Appendix) suggests that cfhc(TSn) might be something near c x 3.5 ~, for some constant c. There are several rectilinear drawings o f Kn that have fewer crossings than TSn, and almost certainly have more cfhc's (see [HI), TS~ was selected for analysis of its number o f cfhc's because its symmetries allow for a recursive counting argument. Crucial to our argument is the fact that any k consecutive vertices of T S , induce a drawing isomorphic to TSk; we know o f no drawing of K. with fewer crossings than TSn which has this property. For all values of n for which cfhc's have been explicitly counted, no drawing of TSn has more cfhc's than a certain non-rectilinear drawing BK, (see [H]); all values of lower bounds for ~ ( n ) which appear in the table in Section 1, correspond to the number o f cfhc's o f BK,. We conjecture that the number of cfhc's o f BK~ serve as an u p p e r bound for both ~ ( n ) and qa(n). From the values o f cfhc (BK~) that appear in this table, we conjecture that cfhc (BK,) is asymptotically c x r~, where 4.3 < r < 4.5. Finally, we conjecture that both ~ ( n ) and ~ ( n ) are less than c x 4.5", for some constant c. Acknowledgments I would like to thank Selim Akl, Jon Davis, David Gregory, and Peter Taylor for the many insightful comments and fruitful suggestions which they offered and which contributed to the development of the ideas in this paper. I am especially indebted to Selim Akl and Peter Taylor, with whom I was in frequent consultation when the original version of this paper was written. Finally, I thank Dave Rappaport and the McGill School of Computer Science Computational Geometry Laboratory for providing the computer facilities used to draw the figures. Appendix This Appendix contains the matrices T6 to T~4 (augmented), the dominant eigenvalues of matrices ,ol to PHRecall that in a cfhc of the drawing TS,, X, Y, and Z represent, respectively, the edges ( n - 2 , n - 1), ( n - 1 , n), and ( n - 2 , n), x, y, and z represent, respectively, the edges (1, 2), (2, 3), and (1, 3). Recall that the entry tn(l'~, a) of matrix T, is the number of cfhc's of the drawing TSn with outermost edge set fl and innermost edge set a, where l) takes on the values X, Y, X, YZ, Z X , X Y and a takes on the values x, y, z, yz, zx, xy. For example, the entry in row 5, column 2 of matrix T7 is the number of cfhc's of TS7 with outermost edge set {Z, X} and innermost edge set {y}, i.e., the number of cfhc's of TS7 that contain edges (5, 7) and (5, 6) but not edge (6, 7), and that contain edge (2, 3) but not edges (1, 2) and (1, 3). For the sake o f completeness, the matrices T, have been augmented in this Appendix to include a seventh row, corresponding to those cfhc's containing none of the edges X, Y, and Z, and a seventh column, corresponding to those cfhc's containing none of the edges x, y, and z. Thus, for n > 3, the sum of all entries of the augmented matrix T, gives the total number of cfhc's of the drawing TSn. Recall that the number of different cfhc's of the drawing TS,+k that can be created by adding exactly k vertices outside a drawing of TS, is given by the equation tn+k >- Nk X tn, (k.1) where Nk = T3+k X Q, and where the entries of the column vector tn are the number of cfhc's of TS, with outermost edge set (respectively) X, Y, Z, YZ, ZX, X Y . The matrix Q is shown below. Recall that the number of different cfhc's of the drawing TS,+k that can be created by adding 1 , 2 , . . . , o r k vertices to the drawings TS,+k_~, TS~+k-2,..., TS,, respectively, is given by the equation Matrices T6 to T14(augmented)--continued With at least one innermost edge and at least one outermost edge [A] S. Akl , A lower bound on the maximum number of crossing-free Hamilton cycles in a rectilinear drawing of K,, Ars Combin. 7 ( 1979 ), 7 - 18 . [ACNS] M. Ajtai , V. Chv~tal, M. Newborn, and E. Szemer6di, Crossing-free subgraphs , Ann. Discrete Math. 12 ( 1982 ), 9 - 12 . [EG] P. Erdos and R. K. Guy , Crossing number problems , Amer. Math. Monthly 80 ( 1973 ), 52 - 58 . [G] R. K. Guy , Unsolved problems , Amer. Math. Monthly 88 ( 1981 ), 757 . [H] R. B. Hayward , The Optimal Crossing-Free Hamilton Cycle Problem for Planar Drawings of the Complete Graph, M.Sc . thesis, Queen's University, Kingston, Ontario, 1982 . [ N M ] M. Newborn and W. O. J. Moser , Optimal crossing-free Hamiltonian circuit drawings of K,, J. Combin. Theory Ser . B 29 ( 1980 ), 13 - 26 .


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Ryan B. Hayward. A lower bound for the optimal crossing-free Hamiltonian cycle problem, Discrete & Computational Geometry, 1987, 327-343, DOI: 10.1007/BF02187887