The Game Saturation Number of a Graph

Given a family F and a host graph H, a graph G ⊆ H is F-saturated relative to H if no subgraph of G lies in F but adding any edge from E(H) - E(G) to G creates such a subgraph. In the F-saturation game on H, players Max and Min alternately add edges of H to G, avoiding subgraphs in F, until G becomes F-saturated relative to H. They aim to maximize or minimize the length of the game, respectively; satg(F;H) denotes the length under optimal play (when Max starts). Let O denote the family of odd cycles and the family of n-vertex trees, and write F for when F = {F}. Our results include satg(O; Kn) = [n/2] [n/2], satg(Tn; Kn) = (n-2/2) + 1 for n ≥ 6, satg(K1,3; Kn) = 2[Tn/2] for n ≥ 8, and satg(P4; Kn) ∈ {[4n/f] , [4n/5]} for n ≥ 5. We also determine satg(P4; Km;n); with m ≥ n, it is n when n is even, m when n is odd and m is even, and m + [n/2] when mn is odd. Finally, we prove the lower bound satg(C4; Kn,n) ≥ 1/21n13/12 – O(n35/36). The results are very similar when Min plays first, except for the P4-saturation game on Km,n.

Extremal/Saturation Numbers for Guessing Numbers of Undirected Graphs

Hat guessing games—logic puzzles where a group of players must try to guess the color of their own hat—have been a fun party game for decades but have become of academic interest to mathematicians and computer scientists in the past 20 years. In 2006, Søren Riis, a computer scientist, introduced a new variant of the hat guessing game as well as an associated graph invariant, the guessing number, that has applications to network coding and circuit complexity. In this thesis, to better understand the nature of the guessing number of undirected graphs we apply the concept of saturation to guessing numbers and investigate the extremal and saturation numbers of guessing numbers. We define and determine the extremal number in terms of edges for the guessing number by using the previously established bound of the guessing number by the chromatic number of the complement. We also use the concept of graph entropy, also developed by Søren Riis, to find a constant bound on the saturation number of the guessing number

Game saturation of intersecting families

We consider the following combinatorial game: two players, Fast and Slow, claim $k$-element subsets of $[n]=\{1,2,...,n\}$ alternately, one at each turn, such that both players are allowed to pick sets that intersect all previously claimed subsets. The game ends when there does not exist any unclaimed $k$-subset that meets all already claimed sets. The score of the game is the number of sets claimed by the two players, the aim of Fast is to keep the score as low as possible, while the aim of Slow is to postpone the game's end as long as possible. The game saturation number is the score of the game when both players play according to an optimal strategy. To be precise we have to distinguish two cases depending on which player takes the first move. Let $gsat_F(\mathbb{I}_{n,k})$ and $gsat_S(\mathbb{I}_{n,k})$ denote the score of the saturation game when both players play according to an optimal strategy and the game starts with Fast's or Slow's move, respectively. We prove that $\Omega_k(n^{k/3-5}) \le gsat_F(\mathbb{I}_{n,k}),gsat_S(\mathbb{I}_{n,k}) \le O_k(n^{k-\sqrt{k}/2})$ holds