Hamiltonian Cycles through Specified Edges in Bipartite Graphs, Domination Game, and the Game of Revolutionaries and Spies by Reza Zamani Nasab

This thesis deals with the following three independent problems. Pósa proved that if G is an n-vertex graph in which any two nonadjacent vertices have degree sum at least n + k, then G has a spanning cycle containing any specified family of disjoint paths with a total of k edges. We consider the analogous problem for a bipartite graph G with n vertices and parts of equal size. Let F be a subgraph of G whose components are nontrivial paths. Let k be the number of edges in F , and let t1 and t2 be the numbers of components of F having odd and even length, respectively. We prove that G has a spanning cycle containing F if any two nonadjacent vertices in opposite partite sets have degree-sum at least n/2+τ(F ), where τ(F ) = dk/2e+ (here = 1 if t1 = 0 or if (t1, t2) ∈ {(1, 0), (2, 0)}, and = 0 otherwise). We show also that this threshold on the degree-sum is sharp when n > 3k. Bostjan Brešar, Sandi Klavžar and Douglas F. Rall proposed a game involving the notion of graph domination number. Two players, Dominator and Staller, occupy vertices of a graph G, playing alternatingly. Dominator starts first. A vertex is valid is to be occupied if adding it to the occupied set enlarges the set of vertices dominated by the occupied set. The game ends when the occupied set becomes a dominating set (A dominating set is a set of vertices U such that every vertex is in U or has a neighbor in U ; the minimum size of a dominating set is the domination number, written γ(G)). Dominator’s goal is to finish the game as soon as possible, and Staller’s goal is to prolong it as much as possible. The size of the dominating set obtained when both players play optimally is the game domination number of G, written as γg(G). The Staller-first game domination number, written as γ ′ g(G), is defined similarly; the only difference is that Staller starts the game. Brešar et al. showed that γ(G) ≤ γg(G) ≤ 2γ(G)− 1 and that for any k and k′ such that k ≤ k′ ≤ 2k − 1, there exists a graph G with γ(G) = k and γg(G) = k ′. Their constructions use graphs with many vertices of degree 1. We present an n-vertex graph G with domination number, minimum degree and connectivity of order θ( √ n) that satisfies γg(G) = 2γ(G) − 1. Building on the work of Brešar et al., Kinnersley proved that |γg(G) − γ′ g(G)| ≤ 1. Brešar et al. defined a