A proof of the 3/4 conjecture for the total domination game

Progress towards the 1/2-Conjecture for the domination game

“…We decided to keep the argument relatively short and simple with the bound 10 17 n + 1 17 , but we believe that with a more careful case analysis the method used in this paper could yield a better constant C < 10 17 in Theorem 1.4. However, it seems that a substantially different approach would be required to improve the bound beyond 7 13 n, as the example G of multiple disjoint copies of the graph G 13 in Figure 1 shows.…”

“…The game transversal number τ g (H) of H is the number of moves played if Edge-hitter starts the game and both players play optimally. In [10], the authors proved the following result. The closed neighbourhood hypergraph of a graph G is defined as hypergraph H G with vertex set V (H G ) = V (G) and hyperedge set E(H G ) = {N G [x]|x ∈ V (G)} consisting of the closed neighbourhoods of vertices in G. As remarked in [3], the domination game played on a graph G can be seen as a special instance of the transversal game being played on the closed neighbourhood hypergraph H G of G. By taking C = {H G : G is a graph of minimum degree at least 2} and c = 1/2, we obtain the following as a direct corollary of Lemma 3.4.…”

“…Theorem 1.4. For every graph G with minimum degree at least 2, we have γ g (G) ≤ 10 17 n + 1 17 . Since 10n/17 + 1/17 < 3n/5 for all n > 5, it is easy to see that Theorem 1.4 confirms Conjecture 1.2.…”