Moshe Rosenfeld

The complexity of finding generalized paths in tournaments

Abstract Given a tournament with n vertices, we consider the number of comparisons needed, in the worst case, to find a permutation υ 1 υ 2 … υ n of the vertices, such that the results of the games υ 1 υ 2 , υ 2 υ 3 ,…, υ n −1 υ n match a prescribed pattern. If the pattern requires all arcs to go forwrd, i.e., υ 1 → υ 2 , υ 2 → υ 3 ,…, υ n −1 → υ n , and the tournament is transitive, then this is essentially the problem of sorting a linearly ordered set. It is well known that the number of comparisons required in this case is at least cn lg n , and we make the observation that O ( n lg n ) comparisons suffice to find such a path in any (not necessarily transitive) tournament. On the other hand, the pattern requiring the arcs to alternate backward-forward-backward, etc., admits an algorithm for which O ( n ) comparisons always suffice. Our main result is the somewhat surprising fact that for various other patterns the complexity (number of comparisons) of finding paths matching the pattern can be cn lg α n for any α between 0 and 1. Thus there is a veritable spectrum of complexities, depending on the prescribed pattern of the desired path. Similar problems on complexities of algorithms for finding Hamiltonian cycles in graphs and directed graphs have been considered by various authors, [2, pp. 142, 148, 149; 4].

Realization of certain generalized paths in tournaments

A tournament T,, consists of a set of n vertices and a single directed edge joining every pair of distinct vertices. We denote the vertices of T’. by {l, . . . , n). A permutation aI, . . . , a, of the vertices is a generalized path. The type af the path is characterized by the sequence a,_1 = e3 l l l E~__~ where si = i-1 if e -+ ai+1 and 6 = -1 if a, -+ ai tl. We also say that mn_l is realized i: T.. In [S], it was conjectured that if it g? 8, then every tournament T, realizes all possible 2*-l types u,._~. Certain types are known to be always realizable. Thus, since every tournament has a Hamiltonian path the types & = +1, -t1, . . . r +1 and (or= 1, -l,..., -1 are always realizable. Gaiinbaum (Harary [3, p. 21 I, ex. 16.261) observed that if n 2 b, then every Tfl has a Hamiltonian path al + 9 l + a, with a, tal. Thus, types irt which all 6 but one have the same sign are realizable. Griinbaum [2] and Rosenfeld [4] proved that every tournament with three exceptions, has an attidirected Hamiltonian path. The exceptions are the regular tournaments TR,, TRS, TR,, when= TR, is the only regulw tournament on 7 vcr’ices with no transitive sub-tournament on 4 vertices. Thus every T,, n a 8 realizes the type o,.__~ = e1 l l l E,_~~ &i = fl)i. Finally, Forcade [l] proved that if n = 2k then every type o,__~ is realizable by T,,. The purpose of this note is to prove that sequences with “larg; blocks” are always realizable.

Edge partitions of the complete symmetric directed graph and related designs

We show that the edges of the complete symmetric directed graph onn vertices can be partitioned into directed cycles (or anti-directed cycles) of lengthn−1 so that any two distinct cycles have exactly one oppositely directed edge in common whenn=pe>3, wherep is a prime ande is a positive integer. When the cycles are anti-directedp must be odd. We then consider the designs which arise from these partitions and investigate their construction.

On Hamilton decompositions of prisms over simple 3-polytopes

In this paper it is shown that certain families of simple 4-polytopes have a Hamilton decomposition, that is, the edges of these polytopes can be partitioned into two Hamilton cycles.

A Result on Hamiltonian Cycles in Generalized Petersen Graphs

The generalized Petersen graph G(n, k), 1≤k≤n−1, is defined as follows: The graph G(n, k) has vertices v0, v1, …, vn−1, v′0, v′1, …, v′n−1 and edges vivi+1, v′iv′i+k and viv′i for all i with 0≤i≤n−1 with all subscripts taken modulo n. In this paper we show that for each k>2 there exists an n(k) such that whenever n≥n(k), then G(n, k) has a Hamiltonian cycle.

Hamiltonian circuits in prisms over certain simple 3-polytopes

In this paper it is shown that the prisms over cyclically 4-connect simple 3-polytopes admit Hamiltonian circuits. It is also shown that if P is a simple 3-polytope all of whose faces are polygons with six sides or less than the prism over P admits a Hamiltonian circuit.

On embedding triangle-free graphs in unit spheres

The study of tiispersed arrangements of points in certain spaces has attracted many authors. lit has produced some intricate constructions and interesting problems. Far a survey of some aspects of this study see 15). In this paper, the study of dispersed arrangements of points has ted to the following conjecture 121: If G is ti triangle-free graph, then there is a function cp : G --+ S’, where Sk denotes the unit sphere in the Euclidean space EL+‘, such that (g, g’)E E(G) if and only if flp(g)- +‘)I] 7> \/3. In this paper. we establish the conjecture for certain families of graphs. in particular, we show that bipartite graphs are embeddable in Sk and obtain bounds on the dimension of the sphere in which a bipartite graph can be embedded.

Parity of Cycles Containing Specified Edges

Publisher Summary Cycles through specified elements in k-connected graphs are a popular topic of investigation. The chapter overviews that in a cubic 2-connected planar graph, any edge that does not belong to a two-edge cut is a chord of an even cycle. It highlights that this result holds in much more general situations (that is, planarity is not needed). The chapter proves that in 3-connected graphs, every pair of edges is almost always contained both in an even and an odd cycle. A full characterization of the exceptional cases is also obtained. Similar results for regular graphs are obtained in the chapter.

Generalized Hamiltonian Paths in Tournaments

Publisher Summary A tournament is an orientation of a complete graph. Any two vertices (players) v, w are adjacent by exactly one arc, either v → w (v beats w) or w → v. Every tournament Tn has a Hamiltonian path—that is, every tournament Tn realizes the sequences 1, . . . , 1 and 0,. . . , 0. Various other binary sequences are known to be always realizable. Certain families of tournaments, most notably tournaments with n = 2k players, realize all possible 2n–1 binary sequences Bn–1. There are n distinct permutations of the players of Tn and only 2n–1 distinct patterns; these facts led the author to conjecture that every tournament Tn, n > 7, realizes every binary sequence Bn–1. This chapter discusses the evolution of this conjecture, the known results and generalizations to oriented trees, general digraphs, orientations of n-chromatic graphs, and complexity of computation.

Are all simple 4-polytopes hamiltonian?

We construct an extensive family of non-Hamiltonian, 4-regular, 4-connected graphs and show that none of these graphs is the graph of a simple 4-polytope.