发表机构
School of Mathematical Sciences, East China Normal University; School of Mathematical Sciences, South China Normal University（华东师范大学数学科学学院; 华南师范大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对Alon提出的关于连通支配数与支配数之差最大值的问题，给出了当k足够大时的渐近精确公式，并证明M(k+1,k)=0，从而解决了该问题。
AI 中文摘要
对于连通图$G$，设$\gamma(G)$和$\gamma_c(G)$分别表示其支配数和连通支配数。设$M(n,k)$为所有最小度至少为$k$的$n$顶点连通图上$\gamma_c(G)-\gamma(G)$的最大值。Alon证明了\\[ 2\left\lfloor\frac{n}{k+1}\right\rfloor-O(1)\le M(n,k)<\frac{n}{k+1} \bigl(\log\lceil\log(k+1)\rceil+3\bigr). \\] 他提出了一个关于在$n-1\ge k\ge 3$时确定或估计$M(n,k)$的问题，并特别指出，缩小上下界之间的$\log\log(k+1)$差距并判定是否$M(n,k)=\Theta\bigl(\frac{n}{k+1}\bigr)$将是有意义的。我们对足够大的$k$给出了Alon问题的渐近答案。更精确地，$M(k+1,k)=0$。对于$n>k+1$，设\\[ \nu=\frac{n}{k+1} \text{ 且 } \Phi(x)= \frac{1}{x\log\\!\frac{x}{x-1}} \\ (\text{对于}x>1). \\] 当$k\to\infty$时，对所有整数$n>k+1$一致地有\\[ M(n,k)= \big(\Phi(\nu)+o(1)\big) \nu\log\log(k+1). \\]
英文摘要
For a connected graph $G$, let $γ(G)$ and $γ_c(G)$ denote its domination number and connected domination number, respectively. Let $M(n,k)$ be the maximum of $γ_c(G)-γ(G)$ over all connected $n$-vertex graphs of minimum degree at least $k$. Alon proved that \[ 2\left\lfloor\frac{n}{k+1}\right\rfloor-O(1)\le M(n,k)<\frac{n}{k+1} \bigl(\log\lceil\log(k+1)\rceil+3\bigr). \] He proposed a problem to determine or estimate $M(n,k)$ for $n-1\ge k\ge 3$, and particularly remarked that it would be interesting to close the $\log\log(k+1)$ gap between the upper and lower bounds and decide whether or not $M(n,k)=Θ\bigl(\frac{n}{k+1}\bigr)$. We give an asymptotic answer to Alon's problem for sufficiently large $k$. More precisely, $M(k+1,k)=0$. For $n>k+1$, let \[ ν=\frac{n}{k+1} \text{ and } Φ(x)= \frac{1}{x\log\!\frac{x}{x-1}} \ (\text{for }x>1). \] As $k\to\infty$, uniformly over all integers $n>k+1$, \[ M(n,k)= \big(Φ(ν)+o(1)\big) ν\log\log(k+1). \]
