$1/2$-conjectures on the domination game and claw-free graphs

Abstract

Let γ g (G) be the game domination number of a graph G. Rall conjectured that if G is a traceable graph, then γ g (G) ≤ 1 2 n(G) . Our main result verifies the conjecture over the class of line graphs. Moreover, in this paper we put forward the conjecture that if δ(G) ≥ 2, then γ g (G) ≤ 1 2 n(G) . We show that both conjectures hold true for claw-free cubic graphs. We further prove the upper bound γ g (G) ≤ 11 20 n(G) over the class of claw-free graphs of minimum degree at least 2. Computer experiments supporting the new conjecture and sharpness examples are also presented.