Tight bounds for generalized power domination in regular graphs

Abstract

Dorbec et al. [SIAM J. Discrete Math., 27 (2013)] conjectured that, for all integers $k\geq1$ and $r\geq3$, every connected $r$-regular graph $G$ of order $n$, other than $K_{r,r}$, satisfies $\gamma_{P,k}(G)\leq n/(r+1)$. After disproving this conjecture, Chen et al.[Graphs Combin., 38 (2022)] proposed a corresponding conjecture for claw-free regular graphs. In this paper, we prove this conjecture: for integers $k\geq\ell\geq1$, every connected claw-free $(k+\ell+1)$-regular graph $G$ of order $n$ satisfies $\gamma_{P,k}(G)\leq n/(k+\ell+2)$, and this bound is tight. Moreover, without the claw-free assumption, we show that, for each fixed integer $k\geq1$, the supremum of $\gamma_{P,k}(G)/\lvert V(G)\rvert$ over all connected $r$-regular graphs $G$ is asymptotic to $(\ln r)/r$ as $r\to\infty$.