In this paper we prove that if T is a regular n-partite tournament with n≥6, then each arc of T lies on a k-cycle for k=4,5,···,n. Our result generalizes theorems due to Alspach and Guo respectively.

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 realize...

The concept of molds, introduced by the authors in a recent preprint, break regular tournaments naturally into big classes: cyclic tournaments, tame tournaments and wild tournaments. We enumerate completely the tame molds, and prove that the dichromatic number of a tame tournament is 3.

In this paper we propose a regularized relaxation based graph matching algorithm. The graph matching problem is formulated as a constrained convex quadratic program, by relaxing the permutation matrix to a doubly stochastic one. To gradually push the doubly stochastic matrix back to a permutation one, a simple weighted concave regular term is added to the convex objective function. The concave ...

The vertex set of a digraph D is denoted by V (D). A c-partite tournament is an orientation of a complete c-partite graph. A digraph D is called cycle complementary if there exist two vertex disjoint cycles C1 and C2 such that V (D) = V (C1) ∪ V (C2), and a multipartite tournament D is called weakly cycle complementary if there exist two vertex disjoint cycles C1 and C2 such that V (C1) ∪ V (C2...

In this paper we propose a new technical tool for analyzing representations of Hilbert C∗-product systems. Using this tool, we give a new proof that every doubly commuting representation over N has a regular isometric dilation, and we also prove sufficient conditions for the existence of a regular isometric dilation of representations over more general subsemigroups of Rk+.

In a recent paper, we showed that every sufficiently large regular digraph G on n vertices whose degree is linear in n and which is a robust outexpander has a decomposition into edge-disjoint Hamilton cycles. The main consequence of this theorem is that every regular tournament on n vertices can be decomposed into (n − 1)/2 edge-disjoint Hamilton cycles, whenever n is sufficiently large. This v...

Doubly and triply N-confused hexaphyrins revealed quite unique photophysical properties arising from confusion of pyrrole rings in the macrocycle, and the molecular shape when compared with their parent regular hexaphyrin molecules.