On a conjecture on total domination in claw-free cubic graphs

In an earlier manuscript [O. Favaron, M.A. Henning, Bounds on total domination in claw-free cubic graphs, Discrete Math. 308 (2008) 3491–3507] it is shown that if GG is a connected claw-free cubic graph of order n≥10n≥10, then γt(G)≤5n/11γt(G)≤5n/11 and it is conjectured that the bound can be improved from 5n/115n/11 to 4n/94n/9. In this paper, we prove this conjecture. Our proof assigns weights to the edges and uses discharging rules to determine the average sum of the edge weights incident to each vertex, and then uses counting arguments to establish the desired upper bound.