We define and study embeddings of cycles in finite affine and projective planes. We show that for all k, 3 ≤ k ≤ q 2 , a k-cycle can be embedded in any affine plane of order q. We also prove a similar result for finite projective planes: for all k, 3 ≤ k ≤ q 2 + q + 1, a k-cycle can be embedded in any projective plane of order q.

Connections are made between deficiency one α-flokki and Baer groups of associated α-flokki translation planes, extending the theory of Johnson and Payne-Thas. The full collineation group of an α-flokki is completely determined. Many of the ideas are extended to the infinite case.

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 K n × 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.

We define a two-player combinatorial game in which players take alternate turns; each turn consists on 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.In this article we define a two-person game played on the vertices of a graph, and then study it to find strategies for either player to win. The analysis of the game ends up depending heavily on the family of graphs considered at the time. This is why, in this article, we will consider a wide variety of tools from game theory, combinatorics and group theory, plus some programming, to attack this problem. In the following section we will cover some basic notation and definitions that will be useful throughout the paper, as well as the actual game play.2000 Mathematics Subject Classification. Primary 05C57, 91A43, 91A46. Secondary 68R10.

We prove that every possible k-cycle can be embedded into P G(n, q), for all n ≥ 3 and q a power of a prime.