## 1 Introduction

Linear arrays and rings are two of the most important computational structures in interconnection networks. So, embedding of linear arrays and rings into a faulty interconnected network is one of the important issues in parallel processing. An interconnection network is often modeled as a graph, in which vertices and edges correspond to nodes and communication links, respectively. Thus, the embedding problem can be modeled as finding fault-free paths and cycles in the graph with some faulty vertices and/or edges. In the embedding problem, if the longest path or cycle is required the problem is closely related to well-known hamiltonian problems in graph theory. In the rest of this paper, we will use standard terminology in graphs(see ref.[2]). It is very difficult to determine that a graph is hamiltonian or not. Readers may refer to [4,5,6].

## 2 Definitions and Notation

We follow [2] for graph-theoretical terminology and notation not defined here. A graph *G* = (*V,E*) always means a simple graph(without loops and multiple edges), where *V* = *V*(*G*) is the vertex set and *E* = *E* (*G*) is the edge set. The *degree* of a vertex *v* is denoted by *d*(*v*) = *d _{G}*(

*v*), whereas

*d*=

*δ*(

*G*) and

*Δ*=

*Δ*(

*G*) stand for the minimum degree and the maximum degree of

*G*, respectively. A

*path*is a sequence of adjacent vertices, denoted by

*x*

_{1}

*x*

_{2}

*· · · x*, in which all the vertices

_{l}*x*

_{1}

*,x*

_{2}

*, · · · ,x*are distinct. Let path

_{l}*P*=

*x*

_{1}

*,x*

_{2}

*, · · · ,x*and

_{‘}*P*[

*x*] denotes the sequence of adjacent vertices from

_{i},x_{j}*x*to

_{i}*x*on path

_{j}*P*. That is,

*P*[

*x*] =

_{i},x_{j}*x*

_{i}x_{i}_{+1}

*· · ·x*

_{j-}_{1}

*x*. For two vertices

_{j}*u,v ϵ V*(

*G*), a path joining

*u*and

*v*is called a

*uv*-path. A path

*P*in graph

*G*is call a Hamiltonian path if

*V*(

*P*) =

*V*(

*G*). A

*cycle*is a path such that the first vertex is the same as the last one. A cycle is also denoted by

*x*

_{1}

*x*

_{2}

*· · · x*

_{l}x_{1}.

*Length*of a cycle is the number of edges in it. An

*‘*-cycle is a cycle of length

*l*. Let

*c*(

*G*) =

*max{l*:

*‘*-cycle of

*G}*. A cycle of

*G*is called

*Hamiltonian cycle*if its length is

*|V*(

*G*)

*|*. A graph

*G*is

*Hamiltonian*if

*G*contains a Hamiltonian cycle. A graph

*G*is

*Hamiltonian-connected*if any two vertices

*u,v 2 V*(

*G*) exists a Hamiltonian

*uv*-path.

Two edges *xy,uv 2 E* (*G*) are called *independent* if *{x,y}∩{u,v}* = / 0. A *matching* is a set of edges that are pairwise independent. A *perfect matching* between two disjoint graphs *G*_{1}, *G*_{2} with the same order *n* is a matching consisting of *n* edges such that each of them has one end vertex in *G*_{1} and the other one in *G*_{2}. See [7, 8, 9, 10, 11, 12, 13, 14].

The construction of new graphs from two given ones is not unusual at all. Basically, the method consists of joining together several copies of one graph according to the structure of another one, the latter being usually called the main graph of the construction. In this regard, Chartrand and Harary introduced in [3] the concept of permutation graph as follows. For a graph *G* and a permutation *π* of *V*(*G*), the permutation graph *G ^{π}* is defined by taking two disjoint copies of

*G*and adding a perfect matching joining each vertex

*v*in the first copy to

*π*(

*v*) in the second. Examples of these graphs include hypercubes, prisms, and some generalized Petersen graphs. The product graph

*G*of two given graphs

_{m}* G_{p}*G*and

_{m}*G*, defined in [1] by Bermond et al. in the following way.

_{p}Definition 1. Let *G _{m}* = (

*V*(

*G*)

_{m}*,E*(

*G*)) and

_{m}*G*= (

_{p}*V*(

*G*)

_{p}*,E*(

*G*)) be two graphs. Let us give an arbitrary orientation to the edges of

_{p}*G*, in such a way that an arc from vertex

_{m}*x*to vertex

*y*is denoted by

*e*. For each arc

_{xy}*e*, let

_{xy}*π*be a permutation of

_{exy}*V*(

*G*). Then the product graph

_{p}*G*has

_{m}*G_{p}*V*(

*G*)

_{m}*×V*(

*G*) as vertex set, with two vertices (

_{p}*x,x*), (

^{'}*y,y*) being adjacent if either

^{'}or

The product graph *G _{m} * G_{p}* can be viewed as formed by

*|V*(

*G*)

_{m}*|*disjoint copies of

*G*, each arc

_{p}*e*, indicating that some perfect matching between the copies

_{xy}*x*and

*y*of

*G*) is added. So the graph

_{m}*G*is usually called the

_{m}*main graph*and

*G*is called the

_{p}*pattern graph*of the product graph

*G*. Moreover, every edge of

_{m}* G_{p}*G*that belongs to any of the

_{m}* G_{p}*|E*(

*G*)

_{m}*|*perfect matchings between copies of

*G*is an

_{p}*cross*edge of

*G*.

_{m}* G_{p}Observe that if we choose *π _{exy}*(

*x*) =

^{'}*x*for any arc

^{'}*e*then

_{xy}*G*=

_{m}* G_{p}*Gm□G*. Furthermore, if

_{p}*G*is

_{m}*K*

_{2}we have

*K*

_{2}

** G*=

*G*, a permutation graph. Hence,

^{*}*G*can be considered as a generalized permutation graph.

_{m}* G_{p}Definition 2. A graph *G* is called *f -*fault hamiltonian (resp. *f* -fault hamiltonian-connected) if there exist a hamiltonian cycle (resp. if each pair of vertices are joined by a hamiltonian path) in *G -F* for any set *F* of faulty elements with *|F| ≤ f*.

If a graph *G* is *f -*fault hamiltonian (resp. *f -*fault hamiltonian-connected), then it is necessary that *f ≤ δ*(*G*)*-*2 (resp. *f ≤ δ*(*G*)*-*3), where *δ*(*G*) is the minimum degree of *G*.

Definition 3. A graph *G* is call *f -*fault *q*-panconnected if each pair of fault-free vertices are joined by a path in *G -F* of every length from *q* to *|V*(*G-F*)*|-*1 inclusive for any set *F* of fault elements with *|F| ≤ f*.

When we are to construct a path from *s* to *t*, *s* and *t* are called a source and a sink, respectively, and both of them are called terminals. When we find a path/cycle, sometimes we regard some fault-free vertices and edges as faulty elements. They are called *virtual* faults.

## 3 Main Results

For the sake of convenience, we write *m*(*p*) for the order of *G _{m}*(

*G*, respectively.) Suppose

_{p}*V*(

*G*) =

_{m}*{*x

*1,x*

_{2}

*, · · · ,x*and

_{m}}*V*(

*G*) =

_{p}*{*y

*1,y*

_{2}

*, · · · ,y*.

_{p}}*If G _{m} is hamiltonian, and G_{p} is hamiltonian-connected, then G_{m} * G_{p} is also hamiltonian-connected*.

*Proof*. Choose two vertices in *u,v ϵ V*(*G _{m} * G_{p}*). We have the following cases:

Case 1. *u,v* belong to the same subgraph, say, *G _{m}* is hamiltonian, suppose hamiltonian-cycle

*C*=

*x*

_{1}

*x*

_{2}

*· · ·x*. Since

_{m}*u,v*are connected by a hamiltonian-path

*P*

_{1}in

*P*

_{1}= (

*x*

_{1}

*,y*

_{1})(

*x*

_{1}

*,y*

_{2})

*· · ·*(

*x*

_{1}

*,y*), where (

_{p}*x*

_{1}

*,y*

_{1}) =

*u*, (

*x*

_{1}

*,y*) =

_{p}*v*. Since

*x*

_{1}

*x*(

_{m}ϵ E*G*), suppose (

_{m}*x*

_{1}

*,y*

_{2})(

*x*

_{m},y_{2})

*ϵ E*(

*G*). Then

_{m}* G_{p}*u,v*are connected by a hamiltonian-path

*P*in

^{'}*G*by choosing the endvertex, say, (

_{m}* G_{p}*x*

_{2}

*,y*

_{1}) of cross edge (

*x*

_{1}

*,y*

_{1})(

*x*

_{2}

*,y*

_{1}) and have a hamiltonian-cycle

*C*

_{2}passing (

*x*

_{2}

*,y*

_{1}) in

*x*

_{2}

*,y*) in

_{p}*C*

_{2}, and choosing the endvertex, say, (

*x*

_{3}

*,y*) of cross edge (

_{p}*x*

_{3}

*,y*)(

_{p}*x*

_{2}

*,y*) and have a hamiltonian-cycle

_{p}*C*

_{3}passing (

*x*

_{3}

*,y*) in

_{p}*x*

_{3}

*,y*) in

_{l}*C*

_{3}, and so on, choosing the endvertex, say, (

*x*) (1

_{m},y_{j}*≤ j ≤ p*) of cross edge (

*x*

_{m}^{-}_{1}

*,y*

_{1})(

*x*) and choosing the endvertex (

_{m},y_{j}*x*

_{m},y_{2}) of cross edge (

*x*

_{1}

*,y*

_{2})(

*x*

_{m},y_{2}), have a hamiltonian-path

*P*passing (

_{m}*x*) and (

_{m},y_{j}*x*

_{m},y_{2}) in

*P*=

^{'}*u*(

*x*

_{2}

*,y*

_{1})

*∪C*

_{2}[(

*x*

_{2}

*,y*

_{1}),(

*x*

_{2}

*,y*)]

_{p}*∪*(

*x*

_{2}

*,y*)(

_{p}*x*

_{3}

*,y*)

_{p}*∪C*

_{3}[(

*x*

_{3}

*,y*),(

_{p}*x*

_{3}

*,y*)]

_{l}*∪· · ·∪*(

*x*

_{m-}_{1}

*,y*

_{1})(

*x*)

_{m},y_{j}*∪ P*[(

_{m}*x*),(

_{m},y_{j}*x*

_{m},y_{2})]

*∪*(

*x*

_{m},y_{2})(

*x*

_{1}

*,y*

_{2})

*∪P*

_{1}[(

*x*

_{1}

*,y*

_{2})

*,v*]. Then we obtain the desired result.

Case 2. *u,v* belong to two subgraphs, suppose *≤ i ≤ m*). and *u* = (*x*_{1}*,y*_{1}), *v* = (*x _{i},y_{j}*) (1

*≤ j ≤ p*) . Since

*G*is hamiltonian, suppose hamiltonian-cycle

_{m}*C*=

*x*

_{1}

*x*

_{2}

*· · ·x*

_{m}x1. Since

*u*belongs to a hamiltonian-cycle

*C*

_{1}in

*C*

_{1}= (

*x*

_{1}

*,y*

_{1})(

*x*

_{1}

*,y*

_{2})

*· · ·*(

*x*

_{1}

*,y*)(

_{p}*x*

_{1}

*,y*

_{1}), where (

*x*

_{1}

*,y*

_{1}) =

*u*.

Then *u,v* are connected by a hamiltonian-path *P ^{'}* in

*G*by choosing the endvertex, say, (

_{m}* G_{p}*x*

_{2}

*,y*

_{1}) and (

*x*

_{2}

*,y*

_{2}) of cross edges (

*x*

_{1}

*,y*

_{1})(

*x*

_{2}

*,y*

_{1}) and (

*x*

_{1}

*,y*

_{2})(

*x*

_{2}

*,y*

_{2}). Then there exist a hamiltonian-path

*P*

_{2}connecting (

*x*

_{2}

*,y*

_{1}) and (

*x*

_{2}

*,y*

_{2}) in

*x*

_{2}

*,y*) and (

_{t}*x*

_{2}

*,y*

_{t}_{+1}) in path

*P*

_{2}, then

*P*

_{2}=

*P*

_{21}

*∪*(

*x*

_{2}

*,y*)(

_{t}*x*

_{2}

*,y*

_{t}_{+1})

*[P*

_{22}, where

*P*

_{21}=

*P*

_{2}[(

*x*

_{2}

*,y*

_{1}), (

*x*

_{2}

*,y*)] and

_{t}*P*

_{22}=

*P*

_{2}[(

*x*

_{2}

*,y*

_{t}_{+1}), (

*x*

_{2}

*,y*

_{2})]. Choosing the endvertices , say, (

*x*

_{3}

*,y*) and (

_{t}*x*

_{3}

*,y*

_{t}_{+1}) of cross edges (

*x*

_{3}

*,y*)(

_{t}*x*

_{2}

*,y*) and (

_{t}*x*

_{3}

*,y*

_{t}_{+1})(

*x*

_{2}

*,y*

_{t}_{+1}). Then there exist a hamiltonian-path

*P*

_{3}connecting (

*x*

_{3}

*,y*) and (

_{t}*x*

_{3}

*,y*

_{t}_{+1}) in

*x*

_{3}

*,y*) and (

_{j}*x*

_{3}

*,y*

_{j}_{+1}) in path

*P*

_{3}, then

*P*

_{3}=

*P*

_{31}

*∪*(

*x*

_{3}

*,y*)(

_{j}*x*

_{3}

*,y*

_{j}_{+1})

*∪ P*

_{32}, where

*P*

_{31}=

*P*

_{3}[(

*x*

_{3}

*,y*), (

_{t}*x*

_{3}

*,y*)] and

_{j}*P*

_{32}=

*P*

_{3}[(

*x*

_{3}

*,y*

_{j}_{+1}), (

*x*

_{3}

*,y*

_{t}_{+1})], and so on. Until we take adjacent vertices (

*x*

_{i-}_{2}

*,y*) and (

_{m}*x*

_{i-}_{2}

*,y*

_{m}_{+1}) in path

*P*

_{i-}_{2}, and the endvertices, say, (

*x*

_{i-}_{1}

*,y*) and (

_{m}*x*

_{i-}_{1}

*,y*

_{m}_{+1}) of cross edges (

*x*

_{i-}_{1}

*,y*)(

_{m}*x*

_{i-}_{2}

*,y*) and (

_{m}*x*

_{i-}_{1}

*,y*

_{m}_{+1})(

*x*

_{i-}_{2}

*,y*

_{m}_{+1}). Then there exist a hamiltonian-path

*P*

_{i-}_{1}connecting (

*x*

_{i-}_{1}

*,y*) and (

_{μ}*x*

_{i-}_{1}

*,y*

_{μ}_{+1}) in

Choosing the endvertex, say, (*x _{m},y_{p}*) of cross edge (

*x*1

*,y*)(

_{p}*x*). Then there exist a hamiltonian-cycle

_{m},y_{p}*C*passing (

_{m}*x*) in

_{m},y_{p}*x*

_{m},y_{1}) in

*C*, taking the endvertex, say, (

_{m}*x*

_{m}-_{1}

*,y*) of cross edge (

_{p}*x*

_{m},y_{1})(

*x*

_{m}-_{1}

*,y*). Then there exist a hamiltonian-cycle

_{p}*C*

_{m}-_{1}passing (

*x*

_{m}-_{1}

*,y*) in

_{p}*x*

_{i}_{+1}

*,y*

_{1}) in

*C*

_{i}_{+1}, taking the endvertex, say, (

*x*) of cross edge (

_{i},y_{p}*x*

_{i}_{+1}

*,y*

_{1})(

*x*).

_{i},y_{p}If (*x _{i},y_{p}*) = (

*x*). Then there exist a hamiltonian-cycle

_{i},y_{j}*C*passing (

_{i}*x*) in

_{i},y_{p}*x*), say, (

_{i},y_{j}*x*

_{i},y_{j-}_{1}) in cycle

*C*and taking the endvertex, say, (

_{i}*x*

_{i}_{+1}

*,y*

_{j-}_{1}) of cross edge (

*x*

_{i}_{+1}

*,y*

_{j-}_{1})(

*x*

_{i},y_{j-}_{1}) (if (

*x*

_{i}_{+1}

*,y*

_{j-}_{1}) = (

*x*

_{i}_{+1}

*,y*), then taking the other adjacent vertex of (

_{p}*x*) such that (

_{i},y_{j}*x*

_{i}_{+1}

*,y*

_{j-}_{1}) ≠ (

*x*

_{i}_{+1}

*,y*)). Then there exist a hamiltonian-path

_{p}*P*

_{i}_{+1}connecting (

*x*

_{i}_{+1}

*,y*) and (

_{p}*x*

_{i}_{+1}

*,y*

_{j-}_{1}) in

If (*x _{i},y_{p}*) ≠ (

*x*). Then there exist a hamiltonian-path

_{i},y_{j}*P*connecting (

_{i}*x*) and (

_{i},y_{p}*x*) in

_{i},y_{j}Let *2*

Corollary 2. *If G _{m} and G_{p} are hamiltonian-connected, then G_{m} * G_{p} is also hamiltonian-connected*.

Corollary 3. *If G _{m} and G_{p} are hamiltonian, then*,

*c*(*G*)_{m}* G_{p}*≥ mp — p*,*There exists a hamiltonian path in G*._{m}* G_{p}

*Proof*. Since *G _{m}* is Hamiltonian, suppose Hamiltonian-cycle

*C*=

^{'}*x*

_{1}

*x*

_{2}

*· · · x*

_{m}x1. Since

*G*is Hamiltonian, suppose Hamiltonian-cycle

_{p}*C*=

^{"}*y*

_{1}

*y*

_{2}

*· · ·y*

_{p}y1.

- We obtain a cycle
*C*as follows: Choosing any vertex, say, (*x*_{1}*,y*_{1}), in . Taking the endvertex, say, ($V({G}_{p}^{{x}_{1}})$ *x*_{2}*,y*_{1}), of cross edge (*x*_{1}*,y*_{1})(*x*_{2}*,y*_{1}) and have a hamiltonian-cycle*C*_{2}passing (*x*_{2}*,y*_{1}) in . Suppose the last vertex is (${G}_{p}^{{x}_{2}}$ *x*_{2}*,y*) in_{p}*C*_{2}. Taking the endvertex, say, (*x*_{3}*,y*_{1}), of cross edge (*x*_{2}*,y*)(_{p}*x*_{3}*,y*_{1}) and have a hamiltonian-cycle*C*_{3}passing (*x*_{3}*,y*_{1}) in The last vertex is (${G}_{p}^{{x}_{3}}$ *x*_{3}*,y*) in_{p}*C*_{3},*· · ·*, and so on, until taking the endvertex, say, (*x*_{m},y_{1}), of cross edge (*x*_{m},y_{1})(*x*_{m-}_{1}*,y*) and have a hamiltonian-cycle_{p}*C*passing (_{m}*x*_{m},y_{1}) in The last vertex is (${G}_{p}^{{x}_{m}}$ *x*) in_{m},y_{p}*C*. Taking the endvertex, say, (_{m}*x*_{1}*,y*), of cross edge (_{j}*x*)(_{m},y_{p}*x*_{1}*,y*) and have a hamiltonian-cycle_{j}*C*_{1}passing (*x*_{1}*,y*) in_{j} . So the length of ((${G}_{p}^{{x}_{1}}$ *x*_{1}*,y*), (_{j}*x*_{1}*,y*_{1}))-path in is at least${G}_{p}^{{x}_{1}}$ , but if ($\lfloor \frac{p}{2}\rfloor $ *x*_{1}*,y*) = (_{j}*x*_{1}*,y*_{1}), then (i) holds. - We obtain a Hamiltonian path as follows: Choosing any vertex, say, (
*x*_{1}*,y*_{1}), in , and have a hamiltonian-cycle$V({G}_{p}^{{x}_{1}})$ *C*_{1}passing (*x*_{1}*,y*_{1}) in The last vertex is (${G}_{p}^{{x}_{1}}$ *x*_{1}*,y*) in_{p}*C*_{1}. Taking the endvertex, say, (*x*_{2}*,y*_{1}), of cross edge (*x*_{1}*,y*)(_{p}*x*_{2}*,y*_{1}) and have a hamiltonian-cycle*C*_{2}passing (*x*_{2}*,y*_{1}) in The last vertex is (${G}_{p}^{{x}_{2}}$ *x*_{2}*,y*) in_{p}*C*_{2}. Taking the endvertex, say, (*x*_{3}*,y*_{1}), of cross edge (*x*_{2}*,y*)(_{p}*x*_{3}*,y*_{1}) and have a hamiltonian-cycle*C*_{3}passing (*x*_{3}*,y*_{1}) in The last vertex is (${G}_{p}^{{x}_{3}}$ *x*_{3}*,y*) in_{p}*C*_{3},*· · ·*, and so on, until taking the endvertex, say, (*x*_{m},y_{1}), of cross edge (*x*_{m},y_{1})(*x*_{m-}_{1}*,y*) and have a hamiltonian-cycle_{p}*C*passing (_{m}*x*_{m},y_{1}) in . The last vertex is (${G}_{p}^{{x}_{m}}$ *x*) in_{m},y_{p}*C*. Then we obtain hamiltonian-path ((_{m}*x*_{1}*,y*_{1}), (*x*))-path._{m},y_{p}$\u25fb$

*If there exists hamiltonian path in G _{m}, G_{p} is hamiltonian, then there exists a hamiltonian path in G_{m} * G_{p}*

*F _{i}* denote the sets of faulty elements in

*F*

_{0}denotes the set of faulty edges in cross edge-set.

*|F*=

_{i}|*f*, 0

_{i}*≤ i ≤ m*. We denote by

*f*the number of faulty vertices in

_{v}*G*, so that

_{m}* G_{p}*G*is

_{m}* G_{p}-F*mp — f*1.

_{v}-*Let G _{m} have a hamiltonian-path, G_{p} is f -fault hamiltonian-connected and f* +1

*-fault hamiltonian, p ≥*3

*f*+6

*, then*,

*for any f ≥*1*, G*+2_{m}* G_{p}is f*-fault hamiltonian, and**for f*= 0*, G*2_{m}* G_{p}with*faulty elements has a hamiltonian cycle unless one faulty element is contained in*${G}_{p}^{{x}_{i}}$ *and the other faulty element is contained in*${G}_{p}^{{x}_{j}},i\ne j,1\le i,j\le m$

*Proof*. (a) Suppose *C* = *x*_{1}*x*_{2} *· · ·x _{m}* is hamiltonian-path in

*G*. Assuming the number of faulty elements

_{m}*|F| ≤ f*+2, we will construct a cycle of length

*l*,

*l*=

*mp — f*, in

_{v}*G*.

_{m}* G_{p}-FCase 1. *f _{i} ≤ f* , 0

*≤ i ≤ m*. Taking two adjacent vertices

*u*,

*G*is

_{p}*f -*fault hamiltonian-connected, there exists hamiltonian

*uv*-path, say,

*P*

_{1}, in

*x,y*) on

*P*

_{1}such that all of

*x̄*, (

*x*, x̄

*x̄*), ȳ, and (

*y*, ȳ) do not belong to

*F*, where

*P*

_{1}and at most

*f*+2 faulty elements can “block” the candidates, at most two candidates per one faulty element. By assumption

*p ≥*3

*f*+6, and the claim is proved. The path

*P*

_{12}can be obtained by merging

*P*

_{1}and a hamiltonian-path

*P*

_{2}in

*x*,

*x̄*), (

*y*,

*ȳ*), of course the edge (

*x,y*) is discarded. Similarly, there exist an edge (

*x*) on

^{'},y^{'}*P*

_{2}such that all of

*F*, where

*P*

_{123}can be obtained by merging

*P*

_{12}and a hamiltonian-path

*P*

_{3}in

*x*) is discarded, and so on, at least we obtain the hamiltonian-path

^{'},y^{'}*P*

_{12· · · m}in

*G*. Therefore

_{m}* G_{P}-F*P*

_{12· · · m}

*∪ uv*is hamiltonian-cycle in

*G*.

_{m}* G_{P}-FCase 2. There exists some *i, j* such that *f _{i}* =

*f*+1,

*f*= 1. Then

_{j}*f*= 0, 0

_{t}*≤ t ≤ m, t*≠

*i, j*. Since

*f ≥*1.

Subcase 2.1. *i* = 0. Taking two adjacent vertices *G _{p}* is

*f -*fault hamiltonian-connected, there exists hamiltonian

*uv*-path, say,

*P*

_{1}, in

*G*is

_{m}* G_{p}*f*+2

*-*fault hamiltonian.

Subcase 2.2. *i* = 1. Since *G _{p}* is

*f*+ 1

*-*fault hamiltonian, Taking hamiltonian cycle in

*G*is

_{m}* G_{p}*f*+2

*-*fault hamiltonian.

Subcase 2.3. *i 6*= 1, 0. Since *G _{p}* is

*f*+1

*-*fault hamiltonian, Taking hamiltonian cycle

*C*in

_{i}*f*=

_{i}*f*+1,

*f*= 1. Then

_{j}*f*= 0, 0

_{t}*≤ t ≤ m, t*≠

*i, j*, then there exist two adjacent edges (

*x,y*), (

*y,z*) on

*C*such that all of

_{i}*x̄*, (

*x*,

*x̄*),

*ȳ*, and (

*y*,

*ȳ*) ,

*z*, and (

^{0}*z, z*), and (

^{'}*y,y*) do not belong to

^{'}*F*, where

*x*, x̄), (

*y*, ȳ), (

*y,y*), (

^{'}*z, z*) are cross edges. Similar to Case 1,

^{'}*P*

_{i-}_{1}connecting the vertex

*y*and

^{'}, z^{'}*P*

_{i}_{+1}connecting the vertex ȳ,

*x̄*. We obtain a cycle

*C*

_{(i-1)i(i+1)}by merging

*C*and hamiltonian path

_{i}*P*

_{i-}_{1}

*,P*

_{i}_{+1}with the edges (

*x, x̄*), (

*y*, ȳ),(

*z, z*), (

^{'}*y,y*), of course the edges (

^{'}*x,y*) and (

*y, z*) are discarded. Taking an edge

*α;β*in

*P*

_{i-}_{1}and

*P*

_{i}_{+1}, respectively. Similar to Case 1, at least we obtain cycle

*C*

_{12· · · m}, then

*G*is

_{m}* G_{p}*f*+2

*-*fault hamiltonian.

Case 3. There exists some *f _{i}* such that

*f*=

_{i}*f*+2. Then

*f*= 0, 0

_{j}*≤ j ≤ m, j*≠

*i*.

Subcase 3.1. *i* = 0. Taking two adjacent vertices *u*,*G _{p}* is

*f -*fault hamiltonian-connected, there exists hamiltonian

*uv*-path, say,

*P*

_{1}, in

*p ≥*3

*f*+6, then there exist an edge (

*x,y*) on

*P*

_{1}such that all of

*ȳ*, (

*x*,

*x̄*), ȳ, and (

*y*, ȳ) do not belong to

*F*, where

*G*is

_{m}* G_{p}*f*+2

*-*fault hamiltonian.

Subcase 3.2. *i* = 1. Since *G _{p}* is

*f*+1

*-*fault hamiltonian, we select an arbitrary faulty element

*a*in

*a*as a virtual fault-free element. Taking hamiltonian cycle

*C*

_{1}in

*a*is a faulty vertex on

*C*

_{1}, let

*x*and

*y*be two vertices on

*C*

_{1}next to

*a*, else if

*C*

_{1}passes through the faulty edge

*a*, let

*x*and

*y*be the endvertices of

*a*. The cycle

*C*

_{12}is obtained by merging

*C*

_{1}

*-α*and a hamiltonian-path in

*x̄*and ȳ with cross edges (

*x*,

*x̄*), (

*y*,

*ȳ*), where

*G*is

_{m}* G_{p}*f*+2

*-*fault hamiltonian.

Subcase 3.3. *i 6*= 1, 0. Since *G _{p}* is

*f*+1

*-*fault hamiltonian, we select an arbitrary faulty element

*a*in

*a*as a virtual fault-free element. Taking hamiltonian cycle

*C*in

_{i}*a*is a faulty vertex on

*C*, let

_{i}*x*and

*y*be two vertices on

*C*next to

_{i}*a*, else if

*C*passes through the faulty edge

_{i}*a*, let

*x*and

*y*be the endvertices of

*a*. Let

*z*be the vertices on

*C*next to

_{i}*y*. The cycle

*C*

_{(i-1)i(i+1)}is obtained by merging

*C*and a hamiltonian-path in

_{i}- α*x̄*and ȳ and a hamiltonian-path in

*z*and

^{'}*y*with cross edges (

^{'}*z, z*), (

^{'}*y,y*), (

^{'}*x*,

*x̄*), (

*y*, ȳ), where

*G*is

_{m}* G_{p}*f*+2

*-*fault hamiltonian.

(b) If *f* = 0. Since *G _{p}* is 0

*-*fault hamiltonian-connected and 1

*-*fault hamiltonian, we have similar proof subcase 3.2 by regarding a faulty element as virtual faulty-free element if the two faulty elements is contained in subgraph

Corollary 6. *Let G _{m}* =

*K*

_{2}

*, G*+1

_{p}is f -fault hamiltonian-connected and f*-fault hamiltonian, p ≥*3

*f*+6

*, then*,

*for any f ≥*1*, K*_{2}** G*+2_{p}is f*-fault hamiltonian*.*for f*= 0*, K*_{2}** G*2_{p}with*faulty elements has a hamiltonian cycle unless one faulty element is contained in*${G}_{p}^{{x}_{1}}$ *and the other faulty element is contained in* .${G}_{p}^{{x}_{2}}$

*Let G _{m} have a hamiltonian-path, G_{p} is f -fault q-panconnected and f* +1

*-fault hamiltonian, p ≥*2

*q*+

*f*+2

*, q ≥*2

*f*+5

*, then*,

*for any f ≥*2*, G*+1_{m}* G_{p}is f*-fault q*+*m-panconnected*,*for f*= 1*, G*+_{m}* G_{p}with 2 faulty elements has a path of every length q*m or more joining s and t unless s and t are contain in the same subgraph*${G}_{p}^{{x}_{i}}$ *and their neighbors are the faulty element in* ,${G}_{p}^{{x}_{j}},i\ne j$ *for f*= 0*, G*+_{m}* G_{p}with 1 faulty element has a path of every length q*m or more joining s and t unless s and t are contained in the same subgraph*${G}_{p}^{{x}_{i}}$ *and one of their neighbors is the faulty element in*${G}_{p}^{{x}_{j}},i\ne j$

*Proof*.

- Suppose
*P*=*x*_{1}*x*_{2}*· · ·x*is a hamiltonian-path of_{m}*G*. Assuming the number of faulty elements_{m}*|F|≤ f*+1, we will construct a path of every length*L*,*q*+*m ≤ L ≤ mp — f*1, in_{v}-*G*joining any pair of vertices_{m}* G_{p}— F*s*and*t*.

Case 1. *f _{i} ≤ f* , 0

*≤ i ≤ m*.

Subcase 1.1. When both *s, t* is contained in *P*_{1} of length *l*_{1} in *s* and *t* for every *P*_{12} that passes through vertices in *x,y*), on *P*_{1} such that all of *x̄*, (*x*, *x̄*), ȳ, and (*y*, ȳ), do not belong to *F*, where *l*_{1} candidate edges on *P*_{1} and at most *f* +1 faulty elements can "block" the candidates, at most two candidates per one faulty element. By assumption *q ≥* 2 *f* +5, and the claim is proved. The path *P*_{12} can be obtained by merging *P*_{1} and a path *P*_{2} in *x̄*, ȳ with the edges (*x*, *x̄*), (*y*, ȳ), of course the edge (*x,y*) is discarded. Let *l*_{2} be the length of *P*_{2}, the length *l*_{12} of *P*_{12} can be anything in the range *s* and *t* such that the length of path can be anything in the range *u,v*), on *P*_{2} such that all of *ū* , (*u*, *ū*), *⊽*, and (*v*, *⊽*), do not belong to *F*, where *P*_{123} can be obtained by merging *P*_{12} and a path *P*_{3} in *ū*, *⊽* with the edges (*u*, *ū*), (*v*, *⊽*), of course the edge (*u,v*) is discarded. Let *l*_{3} be the length of *p*_{3}, the length *l*_{123} of *P*_{123} can be anything in the range *P*_{12· · · m}, and *q ≤ l*_{12· · ·m} *≤ mp — f _{v}-*1.

Subcase 1.2. When both *s, t* is contained in *x,y*), (*u,v*) on *P _{i}* such that all of

*x̄*, (

*x*,

*x̄*), ȳ, and (

*y*, ȳ),

*ū*, (

*u*, ū),

*⊽*, and (

*v*,

*⊽*) do not belong to

*F*, where

*x*,

*x̄*), (

*y*, ȳ), (

*u*,

*ū*), (

*v*,

*⊽*) are four cross edges. Since there are

*l*

_{1}candidate edges on

*P*

_{1}and at most

*f*+1 faulty elements can "block" the candidates, at most two candidates per one faulty element. By assumption

*q ≥*2

*f*+5, and the claim is proved. The path

*P*

_{(i-1)i(i+1)}can be obtained by merging

*P*and two paths

_{i}*P*

_{i-}_{1},

*P*

_{i}_{+1}in

*ū*,

*⊽*with the edges (

*u*,

*ū*), (

*v*,

*⊽*), and

*x̄*, ȳ with the edges (

*x*,

*x̄*), (

*y*, ȳ), of course the edge (

*u,v*), (

*x,y*) is discarded. Similar to Subcase 1.1, it follows that the path

*P*

_{12· · · m}, and

*q ≤ l*

_{12· · · m}

*≤ mp — f*1.

_{v}-Subcase 1.3. When *s* is in *t* is in *G* is *f -*fault hamiltonian-connected, then *f ≤ δ*(*G*)*-*3, where *δ*(*G*) is the minimum degree of *G*, then there exist one adjacent vertex *s* such that the cross edges sequence *T ∉ F*, where *=**P*_{1} of length *l*_{1} in *t* and (*x _{j},y^{j}*) for every

*s*and

*t*can be obtained by merging

*P*

_{1}and the cross edges sequence

*T*with the length

*P*

_{1}for every integer in the range

Case 2. There exists some *f _{i}* such that

*f*=

_{i}*f*+1. Then

*f*= 0, 1

_{j}*≤ j ≤ m, j*≠

*i*. Since

*f ≥*2. Similar to case 1, it follows that the result completes.

It immediately follows from Case 1, where the assumption *f ≥* 2 is never used, that *f* = 0, 1, *G _{m} * G_{p}* with

*f*+1 faulty elements has a path of every length

*q*+

*m*or more joining

*s*and

*t*unless

*s*and

*t*are contained in the same subgraph

*c*) is done. we testify the case (b), note that

*G*is 1-fault

_{p}*q*-panconnected. This completes the proof.

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