Bernardo Llano

Mean value for the matching and dominating polynomial

The mean value of the matching polynomial is computed in the family of all labeled graphs with n vertices. We introduce the dominating polynomial of a graph whose coefficients enumerate the dominating sets for a graph and study some properties of the polynomial. The mean value of this polynomial is determined in a certain special family of bipartite digraphs.

Circulant tournaments of prime order are tight

We say that a tournament is tight if for every proper 3-coloring of its vertex set there is a directed cyclic triangle whose vertices have different colors. In this paper, we prove that all circulant tournaments with a prime number p>=3 of vertices are tight using results relating to the acyclic disconnection of a digraph and theorems of additive number theory.

Infinite families of tight regular tournaments

In this paper, we construct inflnite families of tight regular tournaments. In particular, we prove that two classes of regular tournaments, tame molds and ample tournaments are tight. We exhibit an inflnite family of 3-dichromatic tight tournaments. With this family we positively answer to one case of a conjecture posed by V. Neumann-Lara. Finally, we show that any tournament with a tight mold is also tight.

On the acyclic disconnection of multipartite tournaments

The acyclic disconnection of a digraph D is the maximum number of components that can be obtained by deleting from D the set of arcs of an acyclic subdigraph. We give bounds for the acyclic disconnection of strongly connected bipartite tournaments and of regular bipartite tournaments. For the latter case, we exhibit an infinite family of tournaments with acyclic disconnection equal to 4.

k-colored kernels

We study k-colored kernels in m-colored digraphs. An m-colored digraph D has k-colored kernel if there exists a subset K of its vertices such that (i) from every vertex v@?K there exists an at most k-colored directed path from v to a vertex of K and (ii) for every u,v@?K there does not exist an at most k-colored directed path between them. In this paper, we prove that for every integer k>=2 there exists a (k+1)-colored digraph D without k-colored kernel and if every directed cycle of an m-colored digraph is monochromatic, then it has a k-colored kernel for every positive integer k. We obtain the following results for some generalizations of tournaments: 1. m-colored quasi-transitive and 3-quasi-transitive digraphs have a k-colored kernel for every k>=3 and k>=4, respectively (we conjecture that every m-colored l-quasi-transitive digraph has a k-colored kernel for every k>=l+1), and 2. m-colored locally in-tournament (out-tournament, respectively) digraphs have ak-colored kernel provided that every arc belongs to a directed cycle and every directed cycle is at most k-colored.

On the Maximum Number of Translates in a Point Set

AbstractGiven a finite set P⊆ℝd, called a pattern, tP(n) denotes the maximum number of translated copies of P determined by n points in ℝd. We give the exact value of tP(n) when P is a rational simplex, that is, the points of P are rationally affinely independent. In this case, we prove that tP(n)=n−mr(n), where r is the rational affine dimension of P, and mr(n) is the r -Kruskal–Macaulay function. We note that almost all patterns in ℝd are rational simplices. The function tP(n) is also determined exactly when |P|≤3 or when P has rational affine dimension one and n is large enough. We establish the equivalence of finding tP(n) and the maximum number sR(n) of scaled copies of a suitable pattern R⊆ℝ+ determined by n positive reals. As a consequence, we show that

On a Conjecture of Víctor Neumann-Lara

s_{A_{k}}(n)=n-\varTheta (n^{1-1/\pi(k)})

Destroying longest cycles in graphs and digraphs

, where Ak={1,2,…,k} is an arithmetic progression of size k, and π(k) is the number of primes less than or equal to k.

A Note on the Feedback Arc Set Problem and Acyclic Subdigraphs in Bipartite Tournaments

Abstract We disprove the following conjecture due to Victor Neumann-Lara: for every couple of integers ( r , s ) such that r ≥ s ≥ 2 there is an infinite set of circulant tournaments T such that the dichromatic number and the acyclic disconnection of T are equal to r and s respectively. We show that for every integer s ≥ 2 there exists a sharp lower bound b ( s ) for the dichromatic number r such that for every r b ( s ) there is no circulant tournament T satisfying the conjecture with these parameters. We give an upper bound B ( s ) for the dichromatic number r such that for every r ≥ B ( s ) there exists an infinite set of circulant tournaments for which the conjecture is valid.

The Dichromatic Number of Infinite Families of Circulant Tournaments

In 1978, C. Thomassen proved that in any graph one can destroy all the longest cycles by deleting at most one third of the vertices. We show that for graphs with circumference k ? 8 it suffices to remove at most 1 / k of the vertices. The Petersen graph demonstrates that this result cannot be extended to include k = 9 but we show that in every graph with circumference nine we can destroy all 9-cycles by removing 1 / 5 of the vertices. We consider the analogous problem for digraphs and show that for digraphs with circumference k = 2 , 3 , it suffices to remove 1 / k of the vertices. However this does not hold for k ? 4 .