The $h$-edge tolerable diagnosability of balanced hypercubes

TL;DR

This paper determines the $h$-edge tolerable diagnosability of balanced hypercubes under two models, extending the understanding of fault diagnosis capabilities in multiprocessor systems with faulty links.

Contribution

It completely characterizes the $h$-edge tolerable diagnosability of balanced hypercubes under PMC and MM* models, complementing previous results.

Findings

Exact $h$-edge tolerable diagnosability values for $BH_n$

Traditional diagnosability of $BH_n$ derived from these results

Enhanced understanding of fault diagnosis in multiprocessor systems

Abstract

To measure the fault diagnosis capability of a multiprocessor system with faulty links, Zhu et al. [Theoret. Comput. Sci. 758 (2019) 1--8] introduced the $h$-edge tolerable diagnosability. This kind of diagnosability is a generalization of the concept of traditional diagnosability. In this paper, as complement to the results in [Theoret. Comput. Sci. 760 (2019) 1--14], we completely determine the $h$-edge tolerable diagnosability of balanced hypercubes $BH_n$ under the PMC model and the MM$^*$ model. Thus, the traditional diagnosability of $BH_n$ is also determined.

Tables

Table 1. Table 1:   The h ℎ h -edge tolerable diagnosability of B ​ H n 𝐵 subscript 𝐻 𝑛 BH_{n}

	PMC model		MM∗ model
	$0\le h\le 2n-1$	$h=2n$	$0\le h\le 2n$
$n=1$	$1$	$0$	$0$
$n\ge 2$	$2n-h$		$2n-h$

Equations

$$t_{h}^{e}(G)=\left\{\begin{array}[]{lll}2&\hbox{if $G$ is isomorphic to $G_{8}$ and $h=0$};\\ \\ k-1&\hbox{if $G$ is isomorphic to $G_{k+1,k+1}$ and $h=0$};\\ \\ k-h&\hbox{otherwise}\end{array}\right.$$

$$|F_1\cup F_2|$$

$$|F_2|$$

$$|F_1|+|F_2|$$

$$|F_2|$$

$$t_{h}^{e}(BH_{1})=\left\{\begin{array}[]{lll}1,&\hbox{if $0\leq h\leq 1$};\\ 0,&\hbox{if $h=2$}\end{array}\right.$$

Full text

The $h$-edge tolerable diagnosability of balanced hypercubes

††thanks: M. Xu’s research is supported by the National Natural Science Foundation of China (11571044, 61373021) and the Fundamental Research Funds for the Central Universities.

Min Xu***Corresponding author. E-mail address: xum@bnu.edu.cn (M. Xu). Yulong Wei

School of Mathematical Sciences, Beijing Normal University

Laboratory of Mathematics and Complex Systems, Ministry of Education,

Beijing, 100875, China

Abstract To measure the fault diagnosis capability of a multiprocessor system with faulty links, Zhu et al. [Theoret. Comput. Sci. 758 (2019) 1–8] introduced the $h$-edge tolerable diagnosability. This kind of diagnosability is a generalization of the concept of traditional diagnosability. In this paper, as complement to the results in [Theoret. Comput. Sci. 760 (2019) 1–14], we completely determine the $h$-edge tolerable diagnosability of balanced hypercubes $BH_{n}$ under the PMC model and the MM∗ model. Thus, the traditional diagnosability of $BH_{n}$ is also determined.

Keywords Balanced hypercube; Fault diagnosis; PMC model; MM∗ model

1 Introduction

Processor failure has become an ineluctable event in a large-scale multiprocessor system. Thus, to keep the multiprocessor system performing its functions efficiently and economically, recognizing faulty processors correctly is a task of top priority. The process of recognizing faulty processors in a multiprocessor system is called fault diagnosis, and the diagnosability of a system is the maximum number of faulty processors the system can recognize. Historically, many scholars and researchers proposed different models to investigate fault diagnosis. In 1967, Preparata, Metze and Chien [10] proposed the PMC model for fault diagnosis in multiprocessor systems. Under this model, all adjacent processors of a system can test one another. In 1992, by modifying the MM model [8], Sengupta and Dahbura [11] proposed the MM∗ model assuming that each processor has to test two processors if the processor is adjacent to the latter two processors. Some references related to fault diagnosis studies under the PMC model or MM∗ model can be seen in [2, 4, 5, 6, 7, 9, 12, 13, 15, 14, 16, 18, 19, 20, 21, 22, 24, 25, 26, 23].

In recent literature [26], Zhu et al. introduced the $h$-edge tolerable diagnosability $t^{e}_{h}(G)$ to measure the fault diagnosis capability of a multiprocessor system $G$ with faulty links. This kind of diagnosability is a generalization of the concept of traditional diagnosability. Specifically, $t^{e}_{h}(G)$ is the minimum diagnosability of graphs $G-F_{e}$ which satisfy that $F_{e}\subseteq E(G)$ and $|F_{e}|\leq h$. Note that if a processor $u$ has no fault-free neighbors, it is impossible to determine whether $u$ is faulty or not in the fault diagnosis. Then $t^{e}_{h}(G)=0$ for $h\geq k$, where $G$ is a $k$-regular graph. Hence, a key issue for the $h$-edge tolerable diagnosability of a $k$-regular graph study is the case of $0\leq h\leq k$.

Let $C(G)$ be the maximum number of common neighbors of any two vertices in the graph $G$. Wei and Xu determined the $h$-edge tolerable diagnosabilities of regular graphs as follows.

Theorem 1.1** ([16])**

Let $G=(V,E)$ be a $k$-regular triangle-free graph with $k\geq 2$. If $C(G)\leq k-1$, then $t_{h}^{e}(G)=k-h$ under the PMC model for $0\leq h\leq k$.

Theorem 1.2** ([16])**

Let $G=(V,E)$ be a connected $k$-regular triangle-free graph with $k\geq 3$. If $C(G)\leq k-1$, then

[TABLE]

under the MM∗ model for $0\leq h\leq k$.

In Theorem 1.2, $G_{8}$ is the graph with vertex set $V(G_{8})=\{x_{1},x_{2},\ldots,x_{8}\}$ and edge set $E(G_{8})=\{x_{i}x_{i+1}\mid 1\leq i\leq 7\}\cup\{x_{8}x_{1}\}\cup\{x_{j}x_{j+4}\mid 1\leq j\leq 4\}$, and $G_{k+1,k+1}$ is the graph with vertex set $V(G_{k+1,k+1})=\{x_{1},x_{2},\ldots,x_{k+1},y_{1},y_{2},\ldots,y_{k+1}\}$ and edge set $E(G_{k+1,k+1})=\{x_{i}y_{j}\mid 1\leq i\leq k+1,1\leq j\leq k+1,i\neq j\}$.

In this paper, we are concerned with the fault diagnosis capability analysis of balanced hypercubes $BH_{n}$. The $n$-dimensional balanced hypercube $BH_{n}$, as one of important variants of the well-known hypercubes, was proposed by Wu and Huang [17]. In recent years, $BH_{n}$ has received considerable attention. For example, Yang [20, 21] studied the conditional diagnosability of $BH_{n}$ under the PMC model and the MM∗ model. Gu et al. [5] determined the $1,2$-good-neighbor diagnosability of $BH_{n}$ under the PMC model and the MM∗ model. Lin et al. [7] determined the $1,2,3$-extra conditional fault-diagnosability of $BH_{n}$ under the PMC model. Zhang et al. [24] investigated the $(t,k)$-diagnosability of $BH_{n}$ under the PMC model. Although $BH_{n}$ is a $2n$-regular and triangle-free graph, $C(BH_{n})=2n$. Thus, $BH_{n}$ does not satisfy the conditions of Theorems 1.1 and 1.2. As complement to Theorems 1.1 and 1.2, we establish the $h$-edge tolerable diagnosability of balanced hypercubes $BH_{n}$ under the PMC model and the MM∗ model for $0\leq h\leq 2n$ and $n\geq 1$.

2 Terminology and preliminaries

A graph G=\big{(}V(G),E(G)\big{)} is used to represent a system (or a network), where each vertex of $G$ represents a processor and each edge of $G$ represents a link. The connectivity $\kappa(G)$ is the minimum cardinality of all vertex subsets $S\subseteq V(G)$ satisfying that $G-S$ is disconnected or trivial. The neighborhood $N_{G}(v)$ of a vertex $v$ in $G$ is the set of vertices adjacent to $v$. We refer readers to [1] for terminology and notation unless stated otherwise.

In 1997, Wu and Huang proposed balanced hypercubes $BH_{n}$. We restate the definition of $BH_{n}$ as follows.

Definition 2.1** ([17])**

The $n$-dimensional balanced hypercube $BH_{n}=(V(BH_{n}),E(BH_{n}))$ has vertex set $V(BH_{n})=\{(a_{0},a_{1},\ldots,a_{i},\ldots,a_{n-1})\mid a_{i}\in\{0,1,2,3\},0\leq i\leq n-1\}$. Each vertex $(a_{0},a_{1},\ldots,a_{i-1},a_{i},a_{i+1},\ldots,a_{n-1})$ of $BH_{n}$ has $2n$ neighbors:

(1)

$((a_{0}\pm 1)~{}{\rm mod~{}4},a_{1},\ldots,a_{i-1},a_{i},a_{i+1},\ldots,a_{n-1})$,

(2)

$((a_{0}\pm 1)~{}{\rm mod~{}4},a_{1},\ldots,a_{i-1},(a_{i}+(-1)^{a_{0}})~{}{\rm mod~{}4},a_{i+1},\ldots,a_{n-1})$.

Figure 1 shows $BH_{1}$ and $BH_{2}$.

Some basic but useful properties of $BH_{n}$ are presented as follows.

Lemma 2.2** ([17])**

The balanced hypercube $BH_{n}$ is bipartite and $\kappa(BH_{n})=2n$.

Lemma 2.3** ([20])**

Let $u$ be an arbitrary vertex of $BH_{n}$ for $n\geq 1$. Then, for an arbitrary vertex $v$ of $BH_{n}$, either $|N_{BH_{n}}(u)\cap N_{BH_{n}}(v)|=0$, $|N_{BH_{n}}(u)\cap N_{BH_{n}}(v)|=2$, or $|N_{BH_{n}}(u)\cap N_{BH_{n}}(v)|=2n$. Furthermore, there is exactly one vertex $w$ such that $|N_{BH_{n}}(u)\cap N_{BH_{n}}(w)|=2n$.

Now, we introduce the definition of the traditional diagnosability of a graph.

Definition 2.4** ([3])**

A graph $G=(V,E)$ of $n$ vertices is $t$-diagnosable if all faulty vertices can be detected without replacement, provided that the number of faults does not exceed $t$. The diagnosability $t(G)$ of a graph $G$ is the maximum value of $t$ such that $G$ is $t$-diagnosable.

For any two sets $A$ and $B$, we use $A-B$ to denote a set obtained by removing all elements of $B$ from $A$. The symmetric difference of two sets $F_{1}$ and $F_{2}$ is defined as the set $F_{1}\bigtriangleup F_{2}$ $=(F_{1}-F_{2})\cup(F_{2}-F_{1})$. The following lemmas give necessary and sufficient conditions for a graph to be $t$-diagnosable under the PMC model and the MM∗ model.

Lemma 2.5** ([3])**

*A graph $G=(V,E)$ is $t$-diagnosable under the PMC model if and only if for any two distinct subsets $F_{1}$ and $F_{2}$ of $V$ with $|F_{1}|\leq t$ and $|F_{2}|\leq t$, there exists a test from $V-(F_{1}\cup F_{2})$ to $F_{1}\bigtriangleup F_{2}$ *(see Figure 2 ).

Lemma 2.6** ([11])**

*A graph $G=(V,E)$ is $t$-diagnosable under the MM∗ model if and only if for any two distinct subsets $F_{1}$ and $F_{2}$ of $V$ with $|F_{1}|\leq t$ and $|F_{2}|\leq t$, at least one of the following conditions is satisfied *(see Figure 3 ):

1. (1)

There are two vertices $u,w\in V-(F_{1}\cup F_{2})$ and there is a vertex $v\in F_{1}\bigtriangleup F_{2}$ such that $uv\in E$ and $uw\in E$. 2. (2)

There are two vertices $u,v\in F_{1}-F_{2}$ and there is a vertex $w\in V-(F_{1}\cup F_{2})$ such that $uw\in E$ and $vw\in E$. 3. (3)

There are two vertices $u,v\in F_{2}-F_{1}$ and there is a vertex $w\in V-(F_{1}\cup F_{2})$ such that $uw\in E$ and $vw\in E$.

We call sets $F_{1}$ and $F_{2}$ distinguishable under the PMC (resp. MM∗) model if they satisfy the condition of Lemma 2.5 (resp. at least one of the conditions of Lemma 2.6); otherwise, $F_{1}$ and $F_{2}$ are said to be indistinguishable.

Recently, Zhu et al. introduced the definition of the $h$-edge tolerable diagnosability of graphs as follows.

Definition 2.7** ([26])**

Given a diagnosis model and a graph $G$, $G$ is $h$-edge tolerable $t$-diagnosable under the diagnosis model if for any edge subset $F_{e}$ of $G$ with $|F_{e}|\leq h$, the graph $G-F_{e}$ is $t$-diagnosable under the diagnosis model. The $h$-edge tolerable diagnosability of $G$, denoted as $t_{h}^{e}(G)$, is the maximum integer $t$ such that $G$ is $h$-edge tolerable $t$-diagnosable.

Clearly, $t_{0}^{e}(G)=t(G)$ holds for any graph $G$.

3 Main Results

In this section, we investigate the $h$-edge tolerable diagnosability of a balanced hypercube $BH_{n}$ under the PMC model and the MM∗ model.

Wei and Xu gave an upper bound of the $h$-edge tolerable diagnosability of a $k$-regular graph $G$ under the PMC model and the MM∗ model as follows.

Lemma 3.1** ([16])**

Let $G=(V,E)$ be a $k$-regular graph with $k\geq 2$. Then $t_{h}^{e}(G)\leq k-h$ under the PMC model and the MM∗ model for $0\leq h\leq k$.

Thus, we immediately obtain the upper bound of the $h$-edge tolerable diagnosability of a balanced hypercube $BH_{n}$ under the PMC model and the MM∗ model.

Corollary 3.2

Let $BH_{n}$ be an $n$-dimensional balanced hypercube. Then $t_{h}^{e}(BH_{n})\leq 2n-h$ under the PMC model and MM∗ model for $0\leq h\leq 2n$.

Now, we give a lower bound of the $h$-edge tolerable diagnosability of a balanced hypercube $BH_{n}$ under the MM∗ model. In the following statements, for a vertex subset $A$ of a graph $G$, we use $N_{G}(A)$ to denote the set \big{(}\bigcup_{v\in A}N_{G}(v)\big{)}-A.

Lemma 3.3

Let $BH_{n}$ be an $n$-dimensional balanced hypercube with $n\geq 2$. Then $t_{h}^{e}(BH_{n})\geq 2n-h$ under the MM∗ model for $0\leq h\leq 2n$.

*Proof. *For an arbitrary edge subset $F_{e}\subseteq E(BH_{n})$ with $|F_{e}|\leq h$, suppose that there exist two distinct vertex subsets $F_{1},F_{2}\subseteq V(BH_{n})$ such that $F_{1}$ and $F_{2}$ are indistinguishable in $BH_{n}-F_{e}$ under the MM∗ model. We will prove the lemma by showing that $|F_{1}|\geq 2n-h+1$ or $|F_{2}|\geq 2n-h+1$. If $|F_{1}\cap F_{2}|\geq 2n-h$, then $|F_{1}|\geq 2n-h+1$ or $|F_{2}|\geq 2n-h+1$. Now, we assume that $|F_{1}\cap F_{2}|\leq 2n-h-1$. Our discussion is divided into two cases as follows.

Case 1. For each vertex $u\in F_{1}\bigtriangleup F_{2}$, $N_{BH_{n}-F_{e}}(u)-(F_{1}\cup F_{2})=\emptyset$.

In this case, choose a vertex $x\in F_{1}\bigtriangleup F_{2}$. Then $N_{BH_{n}-F_{e}}(x)\subseteq F_{1}\cup F_{2}$ and $|N_{BH_{n}-F_{e}}(x)|\geq 2n-h>|F_{1}\cap F_{2}|$. Thus, there exists a vertex $y\in F_{1}\bigtriangleup F_{2}$ such that $xy\in E(BH_{n}-F_{e})$. We have $N_{BH_{n}-F_{e}}(\{x,y\})\subseteq F_{1}\cup F_{2}$. By Lemma 2.2, we know that $BH_{n}$ is a bipartite graph. Note that $|N_{BH_{n}-F_{e}}(\{x,y\})|-|F_{1}\cap F_{2}|\geq(4n-2-h)-(2n-h-1)=2n-1\geq 3$ for $n\geq 2$. Then, there exists a star $K_{1,3}\subseteq BH_{n}[F_{1}\bigtriangleup F_{2}]-F_{e}$. Note that $|N_{BH_{n}}(V(K_{1,3}))|\geq(2n-3)+(2n-1)+(2n-2)=6n-6$ (see Figure 4).

Since $N_{BH_{n}-F_{e}}(V(K_{1,3}))\cup V(K_{1,3})\subseteq F_{1}\cup F_{2}$,

[TABLE]

The last inequality holds for $n\geq 2$. Then $|F_{1}|\geq 2n-h+1$ or $|F_{2}|\geq 2n-h+1$.

Case 2. $N_{BH_{n}-F_{e}}(F_{1}\bigtriangleup F_{2})-(F_{1}\cup F_{2})\neq\emptyset$.

Without loss of generality, suppose that $u\in F_{1}-F_{2}$, $v\in V(BH_{n})-(F_{1}\cup F_{2})$ such that $uv\in E(BH_{n}-F_{e})$. Note that $F_{1}$ and $F_{2}$ are indistinguishable in $BH_{n}-F_{e}$ under the MM∗ model. Then $N_{BH_{n}-F_{e}}(v)-\{u\}\subseteq F_{2}$ and $|N_{BH_{n}-F_{e}}(v)\cap(F_{2}-F_{1})|\leq 1$. Thus, $|F_{1}\cap F_{2}|\geq|N_{BH_{n}-F_{e}}(v)|-2\geq 2n-h-2$. If $|F_{1}-F_{2}|\geq 3$ or $|F_{2}-F_{1}|\geq 3$, then $|F_{1}|\geq 2n-h+1$ or $|F_{2}|\geq 2n-h+1$.

Next, we suppose that $|F_{1}-F_{2}|\leq 2$ and $|F_{2}-F_{1}|\leq 2$.

Case 2.1. For each vertex $w\in F_{1}\bigtriangleup F_{2}$, $|N_{BH_{n}-F_{e}}(w)-(F_{1}\cup F_{2})|\leq 1$.

Note that $N_{BH_{n}-F_{e}}(\{u,v\})-(F_{1}-F_{2})\subseteq F_{2}$. Thus,

[TABLE]

for $n\geq 2$. The second inequality holds for $BH_{n}$ is a bipartite graph.

Case 2.2. For some vertex $w\in F_{1}\bigtriangleup F_{2}$, $|N_{BH_{n}-F_{e}}(w)-(F_{1}\cup F_{2})|\geq 2$.

If there exists a vertex subset $\{v_{1},v_{2}\}\subseteq N_{BH_{n}-F_{e}}(w)-(F_{1}\cup F_{2})$ for some vertex $w\in F_{1}\bigtriangleup F_{2}$ such that $N_{BH_{n}}(v_{1})\neq N_{BH_{n}}(v_{2})$, then by Lemma 2.3, $|N_{BH_{n}}(v_{1})\cap N_{BH_{n}}(v_{2})|=2$. Without loss of generality, assume that $w\in F_{1}-F_{2}$. Since $F_{1}$ and $F_{2}$ are indistinguishable in $BH_{n}-F_{e}$ under the MM∗ model, $N_{BH_{n}-F_{e}}(v_{1})-\{w\}\subseteq F_{2}$ and $N_{BH_{n}-F_{e}}(v_{2})-\{w\}\subseteq F_{2}$ (see Figure 5). Thus,

[TABLE]

The last inequality holds for $n\geq 2$.

Otherwise, by Lemma 2.3, for each vertex $w\in F_{1}\bigtriangleup F_{2}$, $|N_{BH_{n}-F_{e}}(w)-(F_{1}\cup F_{2})|\leq 2$. Without loss of generality, assume that $w\in F_{1}-F_{2}$ and let $\{v_{1},v_{2}\}=N_{BH_{n}-F_{e}}(w)-(F_{1}\cup F_{2})$, where $N_{BH_{n}}(v_{1})=N_{BH_{n}}(v_{2})$ (see Figure 6). Then $|F_{1}\cap F_{2}|\geq 2n-2-\left\lfloor\dfrac{h}{2}\right\rfloor$ and $N_{BH_{n}-F_{e}}(w)-\{v_{1},v_{2}\}\subseteq F_{1}\cup F_{2}$. Note that $(N_{BH_{n}-F_{e}}(w)-\{v_{1},v_{2}\})\cap N_{BH_{n}-F_{e}}(v_{1})=\emptyset$ and $|F_{1}\cup F_{2}|\geq|N_{BH_{n}-F_{e}}(\{w,v_{1}\})\cup\{w\}-\{v_{2}\}|\geq(2n-2)+(2n-1)+1-h=4n-h-2$.

Thus,

[TABLE]

Note that $n\geq 2$. If $n\geq 3$ or $1\leq h\leq 2n$, then $|F_{1}|+|F_{2}|\geq 4n-2h+1$ which means $|F_{1}|\geq 2n-h+1$ or $|F_{2}|\geq 2n-h+1$. Now, we assume that $n=2$ and $h=0$.

If $|F_{1}-F_{2}|=1$, then

[TABLE]

If $|F_{1}-F_{2}|=2$ and $|F_{1}\cap F_{2}|\geq 2n-1-\left\lfloor\dfrac{h}{2}\right\rfloor$, then $|F_{1}|\geq 2n-h+1$.

If $|F_{1}-F_{2}|=2$ and $|F_{1}\cap F_{2}|=2n-2-\left\lfloor\dfrac{h}{2}\right\rfloor=2$, then $BH_{2}-(F_{1}\cap F_{2})$ is connected owing to $|F_{1}\cap F_{2}|=2<4=\kappa(BH_{2})$ by Lemma 2.2. By the assumption that $|N_{BH_{n}-F_{e}}(w)-(F_{1}\cup F_{2})|\leq 2$ for each vertex $w$ in $F_{1}\bigtriangleup F_{2}$, we have $|N_{BH_{2}}(F_{1}\bigtriangleup F_{2})-(F_{1}\cup F_{2})|\leq 2\times|F_{1}\bigtriangleup F_{2}|\leq 8$ and so $|V(BH_{2})-(F_{1}\cup F_{2})-N_{BH_{2}}(F_{1}\bigtriangleup F_{2})|=|V(BH_{2})|-(|F_{1}-F_{2}|+|F_{1}\cap F_{2}|+|F_{2}-F_{1}|)-|N_{BH_{2}}(F_{1}\bigtriangleup F_{2})-(F_{1}\cup F_{2})|\geq 2^{4}-(2+2+2)-8\geq 2$. Thus, there exists a vertex $x$ in $V(BH_{2})-(F_{1}\cup F_{2})$ with $x\notin N_{BH_{2}}(F_{1}\bigtriangleup F_{2})$ connected to a vertex in $F_{1}\bigtriangleup F_{2}$ by a path of $BH_{2}-(F_{1}\cap F_{2})$, which contradicts that $F_{1}$ and $F_{2}$ are indistinguishable in $BH_{2}$ under the MM∗ model by Lemma 2.6.

Thus, $t_{h}^{e}(BH_{n})\geq 2n-h$ under the MM∗ model for $0\leq h\leq 2n$ and $n\geq 2$. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Note that under the PMC model, Case 2 of Lemma 3.3 is non-existent and the proof of Case 1 of Lemma 3.3 also holds for $n\geq 2$. Therefore, by Corollary 3.2 and Lemma 3.3, Theorem 3.4 holds.

Theorem 3.4

Let $BH_{n}$ be an $n$-dimensional balanced hypercube with $n\geq 2$. Then $t_{h}^{e}(BH_{n})=2n-h$ under the PMC model and the MM∗ model for $0\leq h\leq 2n$.

Now, we determine the $h$-edge tolerable diagnosability of balanced hypercube $BH_{1}$.

Theorem 3.5

Let $BH_{1}$ be a $1$-dimensional balanced hypercube. Then

[TABLE]

under the PMC model and $t_{h}^{e}(BH_{1})=0$ under the MM∗ model for $0\leq h\leq 2$.

*Proof. *Note that $BH_{1}$ is isomorphic to a cycle with four vertices. Suppose $V(BH_{1})=\{0,1,2,3\}$ and $E(BH_{1})=\{01,12,23,30\}$ (see Figure 1). Since $BH_{1}$ is $2$-regular, $t_{2}^{e}(BH_{1})=0$ under both diagnosis models.

Let $F_{1}=\{0,1\}$, $F_{2}=\{2,3\}$ and $F_{e}=\emptyset$. Then $F_{1}$ and $F_{2}$ are indistinguishable in $BH_{1}-F_{e}$ under the PMC model. Thus, $t_{1}^{e}(BH_{1})\leq 1$ and $t_{0}^{e}(BH_{1})\leq 1$ under the PMC model.

On the other hand, for an arbitrary edge subset $F_{e}\subseteq E(BH_{1})$ with $|F_{e}|\leq 1$, suppose that two distinct vertex subsets $F_{1},F_{2}\subseteq V(BH_{1})$ satisfy that $|F_{1}|\leq 1$ and $|F_{2}|\leq 1$. Then $F_{1}\cap F_{2}=\emptyset$ and $1\leq|F_{1}\bigtriangleup F_{2}|=|F_{1}\cup F_{2}|\leq 2$. Since $BH_{1}-F_{e}$ is connected for $|F_{e}|\leq 1$, $N_{BH_{1}-F_{e}}(F_{1}\bigtriangleup F_{2})=N_{BH_{1}-F_{e}}(F_{1}\cup F_{2})\neq\emptyset$. Hence, there is an edge between $F_{1}\bigtriangleup F_{2}$ and $V(BH_{1}-F_{e})-(F_{1}\cup F_{2})$. By Lemma 2.5, $t_{h}^{e}(BH_{1})\geq 1$ for $0\leq h\leq 1$ under the PMC model.

Let $F_{1}=\{0\}$, $F_{2}=\{2\}$ and $F_{e}=\emptyset$. Then $F_{1}$ and $F_{2}$ are indistinguishable in $BH_{1}-F_{e}$ under the MM∗ model. Thus, $t_{h}^{e}(BH_{1})\leq 0$ for $0\leq h\leq 1$ under the MM∗ model. Hence, $t_{1}^{e}(BH_{1})=t_{0}^{e}(BH_{1})=0$ under the MM∗ model.

This completes the proof of Theorem 3.5. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

4 Conclusions

In this paper, we determine the $h$-edge tolerable diagnosability of balanced hypercubes $BH_{n}$ under the PMC model and the MM∗ model for $0\leq h\leq 2n$ and $n\geq 1$ (see Table 1). In particular, the traditional diagnosability of $BH_{n}$ is determined. Our future research interest is to investigate the $h$-edge tolerable diagnosability of a regular graph with triangles, which will provide a more precise measure for the fault diagnosis capability of a multiprocessor system.