We give a short constructive proof of a theorem of Fisher: every tournament contains a vertex whose second outneighborhood is as large as its ®rst outneighborhood. Moreover, we exhibit two such vertices provided that the tournament has no dominated vertex. The proof makes use of median orders. A second application of median orders is that every tournament of order 2nÿ 2 contains every arboresce...

Bracketology, the art of successfully picking all the winners in the National Collegiate Athletic Association’s (NCAA) annual men’s Division I college basketball championship tournament, has become a favorite national activity. In spite of the challenges and uncertainty faced in this endeavor, patterns exist in how the seeds appear in each round, particularly the later rounds. This paper statis...

A king u in a tournament is a player who beats any other player v directly or indirectly. That is, either u → v (u beats v) or there exists a third player w such that u→ w and w → v. A sorted sequence of kings in a tournament of n players is a sequence of players, S = (u1, u2, . . . , un), such that ui → ui+1 and ui in a king in sub-tournament Tui = {ui ,ui+1, . . . , un} for i = 1,2, . . . , n...

We study biased orientation games, in which the board is the complete graph Kn, and OMaker (oriented maker) and OBreaker (oriented breaker) take turns in directing previously undirected edges of Kn. At the end of the game, the obtained graph is a tournament. OMaker wins if the tournament has some property P and OBreaker wins otherwise. We provide bounds on the bias that is required for OMaker’s...

We examine the size s(n) of a smallest tournament having the arcs of an acyclic tournament on n vertices as a minimum feedback arc set. Using an integer linear programming formulation we obtain lower bounds s(n) ≥ 3n − 2 − blog2 nc or s(n) ≥ 3n − 1 − blog2 nc, depending on the binary expansion of n. When n = 2k − 2t we show that the bounds are tight with s(n) = 3n−2−blog2 nc. One view of this p...

Tournament solutions, i.e., functions that associate with each complete and asymmetric relation on a set of alternatives a non-empty subset of the alternatives, play an important role within social choice theory and the mathematical social sciences at large. Laffond et al. have shown that various tournament solutions satisfy composition-consistency, a structural invariance property based on the...

The ground and electronically excited states of cyclic N(3) (+) are characterized at the equilibrium D(3h) geometry and along the Jahn-Teller distortions. Lowest excited states are derived from single excitations from the doubly degenerate highest occupied molecular orbitals (HOMOs) to the doubly degenerate lowest unoccupied molecular orbitals (LUMOs), which give rise to two exactly and two nea...

A recurrently occurring problem in combinatorics is the need to completely characterize a finite set of finite objects implicitly defined by a set of constraints. For example, one could ask for a list of all possible ways to schedule a football tournament for twelve teams: every team is to play against every other team during an eleven-round tournament, such that every team plays exactly one ga...

Dual system encryption techniques introduced by Waters in Crypto’09 are powerful approaches for constructing fully secure functional encryption (FE) for many predicates. However, there are still some FE for certain predicates to which dual system encryption techniques seem inapplicable, and hence their fully-secure realization remains an important problem. A notable example is FE for regular la...

‎The symmetric doubly stochastic inverse eigenvalue problem (hereafter SDIEP) is to determine the necessary and sufficient conditions for an $n$-tuple $sigma=(1,lambda_{2},lambda_{3},ldots,lambda_{n})in mathbb{R}^{n}$ with $|lambda_{i}|leq 1,~i=1,2,ldots,n$‎, ‎to be the spectrum of an $ntimes n$ symmetric doubly stochastic matrix $A$‎. ‎If there exists an $ntimes n$ symmetric doubly stochastic ...