正则图中广义功率支配的紧界
Tight bounds for generalized power domination in regular graphs
AI总结：
本文证明了无爪正则图的广义功率支配猜想，得到紧界，还给出不考虑无爪假设时该比值的渐近上确界。
AI中文摘要：
Dorbec等人[SIAM J. Discrete Math., 27 (2013)]猜想，对所有整数k≥1和r≥3，除完全二部图K_{r,r}外，每个阶为n的连通r-正则图G满足γ_{P,k}(G)≤n/(r+1)。Chen等人[Graphs Combin., 38 (2022)]在否定该猜想后，针对无爪正则图提出了对应猜想。本文证明了该猜想：对整数k≥ℓ≥1，每个阶为n的连通无爪(k+ℓ+1)-正则图G满足γ_{P,k}(G)≤n/(k+ℓ+2)，且该界是紧的。此外，不考虑无爪假设时，对每个固定整数k≥1，所有连通r-正则图G的γ_{P,k}(G)/|V(G)|的上确界，当r→∞时渐近于(ln r)/r。
英文摘要：
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 $γ_{P,k}(G)\leq n/(r+1)$. 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 $γ_{P,k}(G)\leq n/(k+\ell+2)$, and this bound is tight. Moreover, without the claw-free assumption, we show that, for each fixed integer $k\geq1$, the supremum of $γ_{P,k}(G)/\lvert V(G)\rvert$ over all connected $r$-regular graphs $G$ is asymptotic to $(\ln r)/r$ as $r\to\infty$.
