Small domination-type invariants in random graphs

TL;DR

This paper investigates the behavior of a generalized domination invariant, called the $c$-self domination number, in random graphs, extending classical domination concepts and analyzing their properties for small values of $c$.

Contribution

It introduces the $c$-self domination number as a unifying generalization of domination, total domination, and Roman domination, and studies its behavior in random graphs for small $c$.

Findings

Analyzes the $c$-self domination number in Erdős–Rényi random graphs.

Provides bounds and asymptotic behavior for small $c$ values.

Connects the generalized invariant to classical domination parameters.

Abstract

For $c\in \mathbb{R}^{+}\cup \{\infty \}$ and a graph $G$, a function $f:V(G)\rightarrow \{0,1,c\}$ is called a $c$-self dominating function of $G$ if for every vertex $u\in V(G)$, $f(u)\geq c$ or $\max\{f(v):v\in N_{G}(u)\}\geq 1$ where $N_{G}(u)$ is the neighborhood of $u$ in $G$. The minimum weight $w(f)=\sum _{u\in V(G)}f(u)$ of a $c$-self dominating function $f$ of $G$ is called the $c$-self domination number of $G$. The $c$-self domination concept is a common generalization of three domination-type invariants; (original) domination, total domination and Roman domination. In this paper, we study a behavior of the $c$-self domination number in random graphs for small $c$.

Equations

$${\gamma }^{\frac{t}{s}}(G(n,p))\in \left[\lfloor a_p(n){\rfloor }_t+\frac{1}{t},b_{p,t}(n)\right]\setminus \left\{b_{p,t}(n)-\frac{i}{t}:t-s+1\le i\le t-1\right\}$$

$$\mathrm{Pr}\left({\gamma }^{\frac{t}{s}}(G(n,p))\in \left[\lfloor a_p(n){\rfloor }_t+\frac{1}{t},b_{p,t}(n)\right]\right)\to 1(n\to \infty )$$

$$g(u)=\{\begin{array}{ll}1& (u=u_0)\\ f(u)& (u\ne u_0).\end{array}$$

$$g(u)=\{\begin{array}{ll}\frac{1}{t}& (u\in U)\\ f(u)& (u\notin U).\end{array}$$

$${\gamma }^{\frac{1}{t}}(G)\le w(g)=w(f)-\frac{|U|(s-1)}{t}\le {\gamma }^{\frac{s}{t}}(G)-\frac{2(s-1)}{t}\le \lfloor {\gamma }^{\frac{s}{t}}(G)\rfloor -\frac{s-1}{t}<\lfloor {\gamma }^{\frac{s}{t}}(G)\rfloor ,$$

$${\gamma }^{\frac{t}{s}}(G(n,p))\in \left[\lfloor a_p(n){\rfloor }_t+\frac{1}{t},b_{p,t}(n)\right]$$

$$\mathrm{Pr}({\gamma }^{\frac{s}{t}}(G(n,p))\le b_{p,t}(n))$$

$$\ge \mathrm{Pr}\left(\gamma (G(n,p))\le \lceil a_p(n)+\frac{1}{2t}\rceil \right)$$

$$\to 1(n\to \infty ).$$

$$f(u)=\{\begin{array}{ll}0& (u\in V(G)\setminus (S_1\cup S_2))\\ 1& (u\in S_1)\\ \frac{1}{t}& (u\in S_2)\end{array}$$

$$I_{S_{1},S_{2}}=\begin{cases}1&(\mbox{$(S_{1},S_{2})$ is a $\frac{1}{t}$-self dominating pair of $G(n,p)$})\\ 0&(\mbox{otherwise}).\end{cases}$$

$$\mathrm{Pr}(I_{S_1,S_2}=1)=\prod _{u\in [n]\setminus (S_1\cup S_2)}\mathrm{Pr}(N_G(u)\cap S_1\ne \varnothing )=(1-(1-p{)}^{m_1}{)}^{n-m_1-m_2}.$$

$$E(X_{m_1,m_2})$$

$$=\sum _{(S_1,S_2)\in {\text{SS}}_{m_1,m_2}}\mathrm{Pr}(I_{S_1,S_2}=1)$$

$$=\left(\genfrac{}{}{0ex}{}{n}{m_1}\right)\left(\genfrac{}{}{0ex}{}{n-m_1}{m_2}\right)(1-(1-p{)}^{m_1}{)}^{n-m_1-m_2},$$

$$\frac{n!}{(n-m_1-m_2)!m_1!m_2!}\le n^{m_1+m_2}\cdot \frac{1}{\sqrt{2\pi m_1}{\left(\frac{m_1}{e}\right)}^{m_1}}\cdot \frac{1}{\sqrt{2\pi m_2}{\left(\frac{m_2}{e}\right)}^{m_2}}<(en{)}^{m_1+m_2};$$

$$\frac{n!}{(n-m_1-m_2)!m_1!m_2!}\le n^{m_{3-i}}\cdot \frac{1}{\sqrt{2\pi m_{3-i}}{\left(\frac{m_{3-i}}{e}\right)}^{m_{3-i}}}<(en{)}^{m_{3-i}}=(en{)}^{m_1+m_2}.$$

$$\frac{n!}{(n-m_1-m_2)!m_1!m_2!}<(en{)}^{m_1+m_2}.$$

$$n(1-p{)}^{m_1}=\frac{n(1-p{)}^{L(\frac{n}{L(n)\ln n})}}{(1-p{)}^{a_p(n)-m_1}}=\frac{n\cdot \frac{L(n)\ln n}{n}}{(1-p{)}^{a_p(n)-m_1}}=\frac{L(n)\ln n}{(1-p{)}^{a_p(n)-m_1}}.$$

$$E(X_{m_1,m_2})$$

$$<(en{)}^{m_1+m_2}{\left(e^{-(1-p{)}^{m_1}}\right)}^{n-m_1-m_2}$$

$$=\exp [(m_1+m_2)+(m_1+m_2)\ln n-n(1-p{)}^{m_1}+(m_1+m_2)(1-p{)}^{m_1}]$$

$$\le \exp \left[2(m_1+m_2)+(m_1+m_2)\ln n-\frac{L(n)\ln n}{(1-p{)}^{a_p(n)-m_1}}\right].$$

$$m_1+m_2\le t\left(m_1+\frac{m_2}{t}\right)\le t{a}_p(n)=t(L(n)-L(L(n)\ln n)).$$

$$E(X_{m_1,m_2})$$

$$\le \exp \left[t(L(n)-L(L(n)\ln n))(\ln n+2)-\frac{L(n)\ln n}{(1-p{)}^{a_p(n)-m_1}}\right]$$

$$=\exp \left[\left(t-\frac{1}{(1-p{)}^{a_p(n)-m_1}}\right)L(n)\ln n+t(2L(n)-L(L(n)\ln n)\ln n-2L(L(n)\ln n))\right]$$

$$<\exp [t(2L(n)-L(L(n)\ln n)\ln n)],$$

$$E(X_a)=E\left(\sum _{(m_1,m_2)\in \mathcal{M}(a)}{X}_{m_1,m_2}\right)=\sum _{\begin{array}{c}(m_1,m_2)\in \mathcal{M}(a)\\ 0\le m_1\le a_p(n)-h_0\end{array}}E(X_{m_1,m_2})+\sum _{\begin{array}{c}(m_1,m_2)\in \mathcal{M}(a)\\ a_p(n)-h_0<m_1\le a\end{array}}E(X_{m_1,m_2}).$$

$$\sum _{\begin{array}{c}(m_1,m_2)\in \mathcal{M}(a)\\ 0\le m_1\le a_p(n)-h_0\end{array}}E(X_{m_1,m_2})$$

$$\le a_p(n)\exp [t(2L(n)-L(L(n)\ln n)\ln n)]$$

$$=\exp [\ln a_p(n)+t(2L(n)-L(L(n)\ln n)\ln n)]$$

$$<\exp [\ln L(n)+t(2L(n)-L(L(n)\ln n)\ln n)]$$

$$\to 0(n\to \infty ),$$

$$m_1+m_2=t\left(m_1+\frac{1}{t}{m}_2\right)-(t-1)m_1<t{a}_p(n)-(t-1)(a_p(n)-h_0)=a_p(n)+(t-1)h_0.$$

$$E(X_{m_1,m_2})$$

$$<\exp [(a_p(n)+\alpha )(\ln n+2)-L(n)\ln n]$$

$$=\exp [-L(L(n)\ln n)\ln n+2L(n)-2L(L(n)\ln n)+\alpha \ln n+2\alpha ]$$

Full text

Key words and phrases. Domination number, Random graph, Self domination number, Roman domination number, Differential.

AMS 2010 Mathematics Subject Classification. 05C69, 05C80.

1 Introduction

Throughout this paper, we let $\mathbb{R}^{+}$ and $\mathbb{Z}^{+}$ denote the set of positive numbers and the set of positive integers, respectively. Let $G$ be a graph. Let $V(G)$ and $E(G)$ denote the vertex set and the edge set of $G$, respectively. For a vertex $u\in V(G)$, we let $N_{G}(u)$ denote the neighborhood of $u$ in $G$; thus $N_{G}(u)=\{v\in V(G):uv\in E(G)\}$. A set $S\subseteq V(G)$ is a dominating set (resp. a total dominating set) of $G$ if each vertex in $V(G)\setminus S$ (resp. each vertex in $V(G)$) is adjacent to a vertex in $S$. The minimum size of a dominating set (resp. a total dominating set) of $G$, denoted by $\gamma(G)$ (resp. $\gamma_{t}(G)$), is called the domination number (resp. the total domination number) of $G$. Since a graph $G$ with isolated vertices has no total dominating set, the total domination number has been typically defined for only graphs without isolated vertices. However, in this paper, we define $\gamma_{t}(G)$ as $\gamma_{t}(G)=\infty$ if $G$ has an isolated vertex for convenience. Domination and total domination are important invariants in graph theory because they have many applications for mathematical problems and real problems (see [5, 6, 7]).

The first author [4] recently defined a new domination-type concept as follows: Let $G$ be a graph. For a function $f:V(G)\rightarrow\mathbb{R}^{+}\cup\{0,\infty\}$, the weight $w(f)$ of $f$ is defined by $w(f)=\sum_{u\in V(G)}f(u)$. Let $c\in\mathbb{R}^{+}\cup\{\infty\}$. A function $f:V(G)\rightarrow\mathbb{R}^{+}\cup\{0,\infty\}$ is a $c$-self dominating function (or $c$-SDF) of $G$ if for each $u\in V(G)$, $f(u)\geq c$ or $\max\{f(v):v\in N_{G}(u)\}\geq 1$. Then the following proposition holds.

Proposition 1.1** **(Furuya [4])

Let $c\in\mathbb{R}^{+}\cup\{\infty\}$, and let $G$ be a graph. If $f$ is a $c$-SDF of $G$, then there exists a $c$-SDF $g$ of $G$ such that $w(g)\leq w(f)$ and $g(u)\in\{0,1,c\}$ for all $u\in V(G)$.

It follows from Proposition 1.1 that the minimum weight of a $c$-SDF of $G$ is well-defined. The minimum weight of a $c$-SDF of $G$, denoted by $\gamma^{c}(G)$, is called the $c$-self domination number of $G$. Note that $\gamma^{1}(G)=\gamma(G)$ and $\gamma^{\infty}(G)=\gamma_{t}(G)$ for all graphs $G$ (see [4]). Furthermore, the $\frac{1}{2}$-self domination number is equal to the half of the Roman domination number defined in Subsection 1.1. Thus self domination concept is a common generalization of three well-studied invariants.

In this paper, our main aim is to analyze a behavior of the $c$-self domination number in Erdős-Rényi model random graphs $G(n,p)$ on $[n]:=\{1,2,\ldots,n\}$. For $p\in(0,1)$ and $n\in\mathbb{Z}^{+}\setminus\{1\}$, let $a_{p}(n)=\log_{1/(1-p)}\frac{n}{\log_{1/(1-p)}n\ln n}$. Then the following are known.

Theorem A** **(Wieland and Godbole [9])

For $p\in(0,1)$, $\gamma(G(n,p))\in\{\lfloor a_{p}(n)\rfloor+1,\lfloor a_{p}(n)\rfloor+2\}$ with probability that tend to $1$ as $n\rightarrow\infty$.

Theorem B** **(Bonato and Wang [2])

For $p\in(0,1)$, $\gamma_{t}(G(n,p))\in\{\lfloor a_{p}(n)\rfloor+1,\lfloor a_{p}(n)\rfloor+2\}$ with probability that tend to $1$ as $n\rightarrow\infty$.

Remark 1

Recall that our definition of total domination is not traditional because we define $\gamma_{t}(G)=\infty$ for graphs $G$ with an isolated vertex. Thus, strictly speaking, total domination in Theorem B is different from one in this paper. However, Bonato and Wang [2] indeed proved that $G(n,p)$ has a total dominating set having size $\lfloor a_{p}(n)\rfloor+2$ with probability that tend to $1$ as $n\rightarrow\infty$. Furthermore, since $\gamma(G)\leq\gamma_{t}(G)$ for all graphs $G$, it follows from Theorem A that $G(n,p)$ has no total dominating set having the size $\lfloor a_{p}(n)\rfloor$ with probability that tend to $1$ as $n\rightarrow\infty$. Hence Theorem B holds under our definition.

By the definition of self domination, if $c,c^{\prime}\in\mathbb{R}^{+}\cup\{\infty\}$ satisfy $c\leq c^{\prime}$, then $\gamma^{c}(G)\leq\gamma^{c^{\prime}}(G)$ for all graphs $G$. Here we note that for $c~{}\in(1,\infty)$, the value $\gamma^{c}(G)$ may be a non-integer if $c$ is a non-integer. Thus the following result is obtained as a corollary of Theorems A and B.

Corollary 1.2

For $c\in[1,\infty)$ and $p\in(0,1)$, $\gamma^{c}(G(n,p))\in[\lfloor a_{p}(n)\rfloor+1,\lfloor a_{p}(n)\rfloor+2]$ with probability that tend to $1$ as $n\rightarrow\infty$.

In this paper, we focus on $c$-self domination in the remaining case, that is, the case where $c\in(0,1)$. To state our main result, we extend the floor $\lfloor*\rfloor$. For $t\in\mathbb{Z}^{+}$ and $a\in\mathbb{R}$, let $\lfloor a\rfloor_{t}$ be the largest number in $\{m_{1}+\frac{m_{2}}{t}:m_{1},m_{2}\in\mathbb{Z},~{}m_{1}+\frac{m_{2}}{t}\leq a\}$. Recall that $a_{p}(n)=\log_{1/(1-p)}\frac{n}{\log_{1/(1-p)}n\ln n}$. For $p\in(0,1)$, $t\in\mathbb{Z}^{+}$ and $n\in\mathbb{Z}^{+}\setminus\{1\}$, let $b_{p,t}(n)=\lfloor\lfloor a_{p}(n)\rfloor_{t}+\frac{1}{t}\rfloor+1$. Note that if $\lfloor a_{p}(n)\rfloor_{t}+\frac{1}{t}$ is a non-integer, then $b_{p,t}(n)$ is the smallest integer more than $a_{p}(n)$; if $\lfloor a_{p}(n)\rfloor_{t}+\frac{1}{t}$ is an integer, then $b_{p,t}(n)$ is the second smallest integer more than $a_{p}(n)$. Our main result is the following:

Theorem 1.3

Let $s$ and $t$ be integers with $2\leq s\leq t-1$. Then for $p\in(0,1)$,

[TABLE]

with probability that tend to $1$ as $n\rightarrow\infty$.

Modeling on existing researches, we find a random variable corresponding to $c$-SDFs and calculate its expected value in Section 3. Then we will obtain a weaker result than Theorem 1.3:

[TABLE]

(see Theorem 3.1). The highlight of this paper is Section 4. While many known results for domination-type invariants in random graphs are completed by just calculating of a random variable, we can refine the above weak result to Theorem 1.3 using additional graph-theoretic approach. Note that $b_{p,t}(n)\leq\lfloor a_{p}(n)\rfloor_{t}+\frac{t+1}{t}$ and $\gamma^{\frac{s}{t}}(G)\in\{m_{1}+\frac{m_{2}}{t}:m_{1},m_{2}\in\mathbb{Z}^{+}\cup\{0\}\}$ for all graphs $G$. Thus Theorem 3.1 claims that $\gamma^{\frac{s}{t}}(G(n,p))$ takes at most $t+1$ values with high probability, and Theorem 1.3 improves “at most $t+1$” to “at most $t-s+2$”. In Subsection 1.1, we focus on the Roman domination number an its related topic.

Remark 2

Using similar strategy in Sections 3 and 4, we can estimate $\gamma^{c}(G(n,p))$ even if $c\in(0,1)$ is irrational number. However, it seems to be difficult to describe an optimal formula. On the other hand, we can give the following rough formula (by Theorem 3.1): Let $c\in(0,1)$ be an irrational number. Then for $p\in(0,1)$ and $\varepsilon\in\mathbb{R}^{+}$, ${\rm Pr}(\gamma^{c}(G(n,p))\in(a_{p}(n),a_{p}(n)+1+\varepsilon])\rightarrow 1~{}~{}(n\rightarrow\infty)$.

1.1 Roman domination and differential

A function $f:V(G)\rightarrow\{0,1,2\}$ is a Roman dominating function of $G$ if each vertex $u\in V(G)$ with $f(u)=0$ is adjacent to a vertex $v\in V(G)$ with $f(v)=2$. The minimum weight of a Roman dominating function of $G$, denoted by $\gamma_{R}(G)$, is called the Roman domination number of $G$. Roman domination was introduced by Stewart [8], and was studied by Cockayne et al. [3] in earnest. Since $\gamma_{R}(G)=2\gamma^{\frac{1}{2}}(G)$ for all graphs $G$, we obtain the following result as a corollary of Theorem 1.3.

Corollary 1.4

For $p\in(0,1)$, $\gamma_{R}(G(n,p))\in\left\{2\lfloor a_{p}(n)\rfloor_{2}+i:1\leq i\leq 3\right\}$ with probability that tend to $1$ as $n\rightarrow\infty$.

Roman domination is closely related to another important invariant. The differential of a graph $G$, denoted by $\partial(G)$, is defined as $\partial(G)=\max\{|(\bigcup_{u\in X}N_{G}(u))-X|-|X|:X\subseteq V(G)\}$. The differential has been widely studied because it was motivated from information diffusion in social networks. Recently, Bermudo et al. [1] proved a very useful result that every graph $G$ satisfies $\gamma_{R}(G)+\partial(G)=|V(G)|$. Thus Corollary 1.4 gives the following.

Corollary 1.5

For $p\in(0,1)$, $\partial(G(n,p))\in\left\{n-2\lfloor a_{p}(n)\rfloor_{2}-i:1\leq i\leq 3\right\}$ with probability that tend to $1$ as $n\rightarrow\infty$.

2 Lemmas

In this section, we prepare some lemmas which will be used in our argument. We start with two fundamental lemmas related to the $c$-self domination concept.

Lemma 2.1

Let $a\in\mathbb{R}^{+}$ and $c\in(0,1)$, and let $G$ be a graph of order at least $a$. Then $\gamma^{c}(G)\leq a$ if and only if there exists a $c$-SDF $f:V(G)\rightarrow\{0,1,c\}$ of $G$ such that $a-1<w(f)\leq a$.

Proof.

The “if” part is trivial. Thus it suffices to prove the “only if” part. Suppose that $\gamma^{c}(G)\leq a$. Then by Proposition 1.1, there exists a $c$-SDF $f$ of $G$ such that $w(f)\leq a$ and $f(u)\in\{0,1,c\}$ for all $u\in V(G)$. Choose $f$ so that $w(f)$ is as large as possible. If $w(f)=|V(G)|$, then $w(f)=a$ because $w(f)\leq a\leq|V(G)|=w(f)$, as desired. Thus we may assume that $w(f)<|V(G)|$. Since $c\in(0,1)$, there exists a vertex $u_{0}\in V(G)$ such that $f(u_{0})\in\{0,c\}$. Then the function $g:V(G)\rightarrow\{0,1,c\}$ with

[TABLE]

is a $c$-SDF of $G$ and $w(g)>w(f)$. This together with the maximality of $w(f)$ implies that $a<w(g)\leq w(f)+1$, and so $a-1<w(f)\leq a$. ∎

Lemma 2.2

Let $s$ and $t$ be integers with $2\leq s\leq t-1$. Let $G$ be a graph, and suppose that $\gamma^{\frac{s}{t}}(G)$ is a non-integer and $\gamma^{\frac{s}{t}}(G)\leq\lfloor\gamma^{\frac{s}{t}}(G)\rfloor+\frac{s-1}{t}$. Then $\gamma^{\frac{1}{t}}(G)<\lfloor\gamma^{\frac{s}{t}}(G)\rfloor$.

Proof.

Let $f:V(G)\rightarrow\{0,1,\frac{s}{t}\}$ be an $\frac{s}{t}$-SDF of $G$ with $w(f)=\gamma^{\frac{s}{t}}(G)$, and let $U=\{u\in V(G):f(u)=\frac{s}{t}\}$. Since $\gamma^{\frac{s}{t}}(G)$ is a non-integer, we have $U\neq\emptyset$. If $|U|=1$, then $\gamma^{\frac{s}{t}}(G)=\lfloor\gamma^{\frac{s}{t}}(G)\rfloor+\frac{s}{t}$, which contradicts the second assumption of the lemma. Thus $|U|\geq 2$.

Let $g:V(G)\rightarrow\{0,1,\frac{1}{t}\}$ be the function with

[TABLE]

Then $g$ is a $\frac{1}{t}$-SDF of $G$, and hence

[TABLE]

as desired. ∎

The following lemmas are well-known (or proved by easy argument) in mathematics.

Lemma 2.3* *(Stirling’s formula)

For $n\in\mathbb{Z}^{+}$, $n!\geq\sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}$.

Lemma 2.4

For $x\geq 0$, $1-x\leq e^{-x}$.

3 A crude estimation

In this section, we prove the following theorem which is weaker than Theorem 1.3

Theorem 3.1

Let $s$ and $t$ be integers with $1\leq s\leq t-1$. Then for $p\in(0,1)$,

[TABLE]

with probability that tend to $1$ as $n\rightarrow\infty$.

In [9], Wieland and Godbole implicitly proved the following lemma.

Lemma 3.2* *(Wieland and Godbole [9])

Let $\varepsilon\in\mathbb{R}^{+}$. Then for $p\in(0,1)$, $\gamma(G(n,p))\leq\lceil a_{p}(n)+\varepsilon\rceil$ with probability that tend to $1$ as $n\rightarrow\infty$.

Lemma 3.3

For $p\in(0,1)$, $t\in\mathbb{Z}^{+}$ and $n\in\mathbb{Z}^{+}\setminus\{1\}$, we have $\lceil a_{p}(n)+\frac{1}{2t}\rceil\leq b_{p,t}(n)$.

Proof.

There exist non-negative integers $m_{1}$ and $m_{2}$ such that $m_{1}+\frac{m_{2}}{t}\leq a_{p}(n)<m_{1}+\frac{m_{2}+1}{t}$ and $0\leq m_{2}\leq t-1$.

Suppose that $m_{2}=t-1$. Since $\lfloor a_{p}(n)\rfloor_{t}+\frac{1}{t}=m_{1}+\frac{t-1}{t}+\frac{1}{t}=m_{1}+1~{}(\in\mathbb{Z}^{+})$, we have $b_{p,t}(n)=\lfloor\lfloor a_{p}(n)\rfloor_{t}+\frac{1}{t}\rfloor+1=m_{1}+2$. On the other hand, $a_{p}(n)+\frac{1}{2t}<m_{1}+1+\frac{1}{2t}$, and so $\lceil a_{p}(n)+\frac{1}{2t}\rceil\leq m_{1}+2=b_{p,t}(n)$, as desired. Thus we may assume that $0\leq m_{2}\leq t-2$.

Since $\lfloor a_{p}(n)\rfloor_{t}+\frac{1}{t}=m_{1}+\frac{m_{2}+1}{t}\leq m_{1}+\frac{t-1}{t}$, we have $b_{p,t}(n)=\lfloor\lfloor a_{p}(n)\rfloor_{t}+\frac{1}{t}\rfloor+1=m_{1}+1$. On the other hand, $a_{p}(n)+\frac{1}{2t}<m_{1}+\frac{t-1}{t}+\frac{1}{2t}=m_{1}+\frac{2t-1}{2t}<m_{1}+1$, and so $\lceil a_{p}(n)+\frac{1}{2t}\rceil\leq m_{1}+1=b_{p,t}(n)$, as desired. ∎

Proof of Theorem 3.1. Note that $\gamma^{\frac{s}{t}}(G)\leq\gamma^{1}(G)=\gamma(G)$ for all graphs $G$. Hence by Lemma 3.2 with $\varepsilon=\frac{1}{2t}$ and Lemma 3.3,

[TABLE]

Consequently, we obtain the upper bound of the theorem.

We next prove the lower bound of the theorem. Let $\mathbb{M}=\{m_{1}+\frac{m_{2}}{t}:m_{1},m_{2}\in\mathbb{Z}^{+}\cup\{0\}\}$, and for $a\in\mathbb{R}^{+}$, let ${\mathcal{M}}(a)=\{(m_{1},m_{2}):m_{1}+\frac{m_{2}}{t}=a\}$. Then ${\mathcal{M}}(a)\neq\emptyset$ if and only if $a\in\mathbb{M}$. Furthermore, we note that $\lfloor a_{p}(n)\rfloor_{t}+\frac{1}{t}$ is the smallest number in $\mathbb{M}$ more than $a_{p}(n)$. Since $\gamma^{\frac{s}{t}}(G)\geq\gamma^{\frac{1}{t}}(G)$ for all graphs $G$, it suffices to show that $\gamma^{\frac{1}{t}}(G(n,p))>a_{p}(n)$ with probability that tend to $1$ as $n\rightarrow\infty$.

For $m_{1},m_{2}\in\mathbb{Z}^{+}\cup\{0\}$, let $X_{m_{1},m_{2}}$ be the random variable counting the number of $\frac{1}{t}$-SDFs $f:[n]\rightarrow\{0,1,\frac{1}{t}\}$ of $G(n,p)$ with $|\{u\in[n]:f(u)=1\}|=m_{1}$ and $|\{u\in[n]:f(u)=\frac{1}{t}\}|=m_{2}$. For $a\in\mathbb{M}$, let $X_{a}=\sum_{(m_{1},m_{2})\in{\mathcal{M}}(a)}X_{m_{1},m_{2}}$.

For a graph $G$, an ordered pair $(S_{1},S_{2})$ of subsets of $V(G)$ with $S_{1}\cap S_{2}=\emptyset$ is called a $\frac{1}{t}$-self dominating pair of $G$ if the function $f:V(G)\rightarrow\{0,1,\frac{1}{t}\}$ with

[TABLE]

is a $\frac{1}{t}$-SDF of $G$. Let $\SS_{m_{1},m_{2}}=\left\{(S_{1},S_{2})\in\binom{[n]}{m_{1}}\times\binom{[n]}{m_{2}}:S_{1}\cap S_{2}=\emptyset\right\}$, and for $(S_{1},S_{2})\in\SS_{m_{1},m_{2}}$, let $I_{S_{1},S_{2}}$ be the random variable satisfying

[TABLE]

Note that $X_{m_{1},m_{2}}=\sum_{(S_{1},S_{2})\in\SS_{m_{1},m_{2}}}I_{S_{1},S_{2}}$. The following claim plays a key role in our argument.

Claim 3.1

For non-negative integers $m_{1}$ and $m_{2}$, $E(X_{m_{1},m_{2}})=\frac{n!}{(n-m_{1}-m_{2})!~{}m_{1}!~{}m_{2}!}(1-(1-p)^{m_{1}})^{n-m_{1}-m_{2}}$.

Proof.

For $(S_{1},S_{2})\in\SS_{m_{1},m_{2}}$, since ${\rm Pr}(N_{G}(u)\cap S_{1}\neq\emptyset)=1-(1-p)^{m_{1}}$ for each $u\in[n]\setminus(S_{1}\cup S_{2})$,

[TABLE]

Since $X_{m_{1},m_{2}}=\sum_{(S_{1},S_{2})\in\SS_{m_{1},m_{2}}}I_{S_{1},S_{2}}$, it follows that

[TABLE]

as desired. ∎

Since $\frac{1}{1-p}>1$, the value $h_{0}=\min\{h\in\mathbb{Z}^{+}:t-\frac{1}{(1-p)^{a}}<0$ for all $a\geq h\}$ is a well-defined constant (depending on $p$ and $t$ only). In the rest of this proof, we consider $G(n,p)$ for sufficiently large $n$. Thus, for example, we may assume that $L(L(n))>0$, $n>ta_{p}(n)$, $a_{p}(n)>h_{0}$, etc. For $x\in\mathbb{R}^{+}$, let $L(x)=\log_{1/(1-p)}x$. Note that $a_{p}(n)=\log_{1/(1-p)}\frac{n}{\log_{1/(1-p)}n\ln n}=L(\frac{n}{L(n)\ln n})$.

Claim 3.2

Let $m_{1}$ and $m_{2}$ be non-negative integers with $a_{p}(n)-1<m_{1}+\frac{m_{2}}{t}\leq a_{p}(n)$. Then the following hold.

1. (i)

We have $E(X_{m_{1},m_{2}})<\exp\left[(m_{1}+m_{2})(\ln n+2)-\frac{L(n)\ln n}{(1-p)^{a_{p}(n)-m_{1}}}\right]$. 2. (ii)

If $0\leq m_{1}\leq a_{p}(n)-h_{0}$, then $E(X_{m_{1},m_{2}})<\exp[t(2L(n)-L(L(n)\ln n)\ln n)]$.

Proof.

1. (i)

By Lemma 2.3, if $m_{1}\geq 1$ and $m_{2}\geq 1$, then

[TABLE]

if $m_{i}=0$ for some $i\in\{1,2\}$, then $m_{3-i}\geq 1$, and hence

[TABLE]

In either case,

[TABLE]

Furthermore, we have

[TABLE]

By Claim 3.1, Lemma 2.4, (3.1) and (3.2),

[TABLE] 2. (ii)

By the definition of $m_{1}$ and $m_{2}$, we have

[TABLE]

Since $a_{p}(n)-m_{1}\geq h_{0}$, it follows from the definition of $h_{0}$ that $(t-\frac{1}{(1-p)^{a_{p}(n)-m_{1}}})L(n)\ln n<0$. This together with (i) and (3.3) implies that

[TABLE]

as desired. ∎

Claim 3.3

Let $a\in\mathbb{M}$ be a number with $a_{p}(n)-1<a\leq a_{p}(n)$. Then $E(X_{a})\rightarrow 0$ if $n\rightarrow\infty$.

Proof.

By the definition of $X_{a}$,

[TABLE]

Note that the number of $m_{1}\in\mathbb{Z}^{+}$ satisfying $a_{p}(n)-h_{0}<m_{1}\leq a$ is at most $h_{0}$ because $a\leq a_{p}(n)$. Hence $\sum_{\begin{subarray}{c}(m_{1},m_{2})\in{\mathcal{M}}(a)\\ a_{p}(n)-h_{0}<m_{1}\leq a\end{subarray}}E(X_{m_{1},m_{2}})$ is a sum having constant terms. Thus it suffices to prove the following:

1. (A1)

$\sum_{\begin{subarray}{c}(m_{1},m_{2})\in{\mathcal{M}}(a)\\ 0\leq m_{1}\leq a_{p}(n)-h_{0}\end{subarray}}E(X_{m_{1},m_{2}})\rightarrow 0~{}~{}(n\rightarrow\infty)$, and 2. (A2)

for each $(m_{1},m_{2})\in{\mathcal{M}}(a)$, if $a_{p}(n)-h_{0}<m_{1}\leq a$, then $E(X_{m_{1},m_{2}})\rightarrow 0~{}~{}(n\rightarrow\infty)$.

By Claim 3.2(ii),

[TABLE]

which proves (A1).

We next assume that $(m_{1},m_{2})\in{\mathcal{M}}(a)$ satisfies $a_{p}(n)-h_{0}<m_{1}\leq a$ and prove (A2). We have

[TABLE]

Note that $\alpha:=(t-1)h_{0}$ is a constant depending on $p$ and $t$ only. Hence it follows from Claim 3.2(i) that

[TABLE]

which proves (A2). ∎

Let $A_{n}=\{a\in\mathbb{M}:a_{p}(n)-1<a\leq a_{p}(n)\}$. Then $|A_{n}|\leq t$. In particular, $\sum_{a\in A_{n}}E(X_{a})$ is a sum having constant terms. Consequently, it follows from Lemma 2.1 and Claim 3.3 that

[TABLE]

and so ${\rm Pr}(\gamma^{\frac{1}{t}}(G(n,p))>a_{p}(n))\rightarrow 1~{}~{}(n\rightarrow\infty)$.

This completes the proof of Theorem 3.1. ∎

4 Graph-theoretical refinement of Theorem 3.1

In this section, we complete the proof of Theorem 1.3. Let $s$, $t$ and $p$ be numbers as in Theorem 1.3. Let $\varepsilon\in\mathbb{R}^{+}$. Then by Theorem 3.1, there exists $N_{0}\in\mathbb{Z}^{+}$ such that for every integer $n\geq N_{0}$,

[TABLE]

Fix an integer $n\geq N_{0}$, and let $i$ be an integer with $t-s+1\leq i\leq t-1$. Since $b_{p,t}(n)$ is an integer, $b_{p,t}(n)-\frac{i}{t}$ is a non-integer. Furthermore, if a graph $G$ satisfies $\gamma^{\frac{s}{t}}(G)=b_{p,t}(n)-\frac{i}{t}$, then

[TABLE]

and hence

[TABLE]

This together with Lemma 2.2 implies that if $\gamma^{\frac{s}{t}}(G)=b_{p,t}(n)-\frac{i}{t}$, then $\gamma^{\frac{1}{t}}(G)<\lfloor\gamma^{\frac{s}{t}}(G)\rfloor=b_{p,t}(n)-1$. Hence we have ${\rm Pr}(\gamma^{\frac{1}{t}}(G(n,p))<b_{p,t}(n)-1)\geq{\rm Pr}(\gamma^{\frac{s}{t}}(G(n,p))=b_{p,t}(n)-\frac{i}{t})$. On the other hand, since $b_{p,t}(n)-1=\lfloor\lfloor a_{p}(n)\rfloor_{t}+\frac{1}{t}\rfloor\leq\lfloor a_{p}(n)\rfloor_{t}+\frac{1}{t}$,

[TABLE]

Consequently,

[TABLE]

and so

[TABLE]

Since $\varepsilon$ is arbitrary, this completes the proof of Theorem 1.3.

Acknowledgment

The authors would like to thank Professor Yoshimi Egawa for his helpful comments about Section 4. This work was partially supported by JSPS KAKENHI Grant number JP18K13449 (to M.F).