On a conjecture on total domination in claw-free cubic graphs: proof and new upper bound

In 2008, Favaron and Henning proved that if G is a connected claw-free cubic graph of order n ≥ 10, then the total domination number γt(G) of G is at most 5 11 n, and they conjectured that in fact γt(G) is at most 4 9 n (see [O. Favaron and M.A. Henning, Discrete Math. 308 (2008), 3491–3507] and [M.A. Henning, Discrete Math. 309 (2009), 32–63]). In this paper, in a first step, we prove this conjecture and show that the bound is reached for exactly two graphs of order 18. In a second step, we prove that if G is a connected claw-free cubic graph of order n ≥ 20, then γt(G) ≤ 10 23n, and we show that this second bound is not reached. Henning and Southey (see [Discrete Math. 310 (2010), 2984–2999] also proved the initial conjecture, but in a less natural way. Moreover, they gave two graphs for which the bound is reached without proving that there are no others. An open problem is proposed in the last section.