Oscar Vega

The Well-Covered Dimension Of Products Of Graphs

Abstract We discuss how to find the well-covered dimension of a graph that is the Cartesian product of paths, cycles, complete graphs, and other simple graphs. Also, a bound for the well-covered dimension of Kn × G is found, provided that G has a largest greedy independent decomposition of length c < n. Formulae to find the well-covered dimension of graphs obtained by vertex blowups on a known graph, and to the lexicographic product of two known graphs are also given.

Combinatorial Analysis of a Subtraction Game on Graphs

We define a two-player combinatorial game in which players take alternate turns; each turn consists of deleting a vertex of a graph, together with all the edges containing such vertex. If any vertex became isolated by a player’s move then it would also be deleted. A player wins the game when the other player has no moves available. We study this game under various viewpoints: by finding specific strategies for certain families of graphs, through using properties of a graph’s automorphism group, by writing a program to look at Sprague-Grundy numbers, and by studying the game when played on random graphs. When analyzing Grim played on paths, using the Sprague-Grundy function, we find a connection to a standing open question about Octal games.

Feet in orthogonal-Buekenhout–Metz unitals

Abstract Given an orthogonal-Buekenhout–Metz unital Uα,β, embedded in PG(2, q2), and a point P ∉ Uα,β, we study the set τP(Uα,β) of feet of P in Uα,β. We characterize geometrically each of these sets as either q + 1 collinear points or as q + 1 points partitioned into two arcs. Other results about the geometry of these sets are also given.

On the Levi graph of point-line configurations

We prove that the well-covered dimension of the Levi graph of a point-line configuration (v_r, b_k) is equal to 0, whenever r > 2.