## 3 Mathematical Preliminaries  
Communication networks are usually illustrated by graphs, in which nodes represent processors and edges represent links between processors. Let \(G=(V,E)\) be a graph if \(V\) is a finite set and \(E\) is a subset of \(\{(a,b)\mid(a,b)\) is an unordered pair of \(V\}\). A path is delimited by \((x_{0},x_{1},x_{2},...x_{n-1})\). A path is called a Hamiltonian path if its nodes are distinct and span \(V\). A cycle is a path of at least three nodes such that the first node is the same as the last node. A cycle is called a Hamiltonian cycle or Hamiltonian if its nodes are distinct except for the first node and the last node, and if they span \(V\)[10]. A bipartite graph \(G=(V,E)\) is a graph such that \(V=A\cup B\) and \(E\) is a subset of \(\{(a,b)\mid a\in A\) and \(b\in B\}\); if \(G\)-\(F\) remains Hamiltonian for any \(F=\{a,b\}\) with \(a\in A\) and \(b\in B\), then \(G\) is \(I_{p}\)-Hamiltonian. A graph \(G\) is \(I\)-edge Hamiltonian if \(G\)-\(e\) is Hamiltonian for any \(e\in E\); moreover, if there is a Hamiltonian path between any pair of nodes \(\{c,d\}\) with \(c\in A\) and \(d\in B\), then the bipartite graph \(G\) is Hamiltonian laceable. The number of links connecting a node is called the _degree_; a network that regularly has fewer degrees is generally economical [11]. For DSRC's functionality in busy traffic conditions, regular degree-3 is minimal for supporting availability, reliability, and maintainability in paths of multi-lanes; consequently, networks prototyped for DSRC are regular degree-3 based in this paper. Assume that \(m\) and \(n\) are positive integers, where \(n\) is even and \(m\geq 2\). Let \(d\) be any integer such that \((m\ -\ d)\) is even. The _generalized honeycomb torus_[12], _GHT_(\(m\), \(n\), \(d\)) is the graph with the node set \(\{(i,j)\mid 0\leq i<m,\,0\leq j<n\}\) such that \((i,j)\) and \((k,l)\) are adjacent if they satisfy one of the following conditions: (1) \(i=k\) and \(j=l\pm 1(\mathbf{mod}\ n)\); (2) \(j=l\) and \(k=i\ -\ 1\) if \(i+j\) is even; and (3) \(i=0\), \(k=m\ -\ 1\), and \(l=j\ +\ d(\mathbf{mod}\ n)\) if \(j\) is even. The spider-web network, _SW_(\(m,n\)) is a graph with the node set \(\{(i,j)\mid 0\leq i<m,\,0\leq j<n\}\), where \(m\), \(n\geq 4\) and even, such that \((i,j)\) and \((k,\,l)\) are adjacent if they satisfy one of the following conditions: (1) \(i=k\) and \(j=l\pm 1\); (2) \(j=l\) and \(k=i\ +\ 1(\mathbf{mod}\ m)\) if \(i+j\) is odd or \(j=n\ -\ 1\); and (3) \(j=l\), \(k=i\ -\ 1(\mathbf{mod}\ m)\) if \(i+j\) is even or \(j=0\). The configuration of _SW_(\(m,n\)) is shown in Figure 1(a,d). It is proved that GHT(\(m\), \(n\), \(0\)), also named _HReT_(\(m\), \(n\)), is _1-edge Hamiltonian_ if \(n\geq 4\); \(I_{p}\)-Hamiltonian if \(n\geq 6\) or \(m=2\), \(n\geq 4\)[13]. When \(m\) and \(n\) are positive integers with \(n\), \(m\ -\ n/2\) being even, _GHT_(\(m\), \(n\), \(n/2\)) is proved _1-edge Hamiltonian_ if \(n\geq 4\); \(I_{p}\)-Hamiltonian if \(n\geq 6\) or \(m=2\), \(n\geq 4\)[14]. Besides, _SW_(\(m\), \(n\)) is proved _1-edge Hamiltonian_, \(I_{p}\)-Hamiltonian [15]. Thus, the fault-tolerance in which we are engaged is systematically based. Moreover, _GHT_(\(m\), \(n\), \(0\)), _GHT_(\(m\), \(n\), \(n\)/2), and _SW_(\(m\), \(n\)) are _Hamiltonian laceable_ if \(m\), \(n\geq 4\) integers [16,17]. Two Hamiltonian paths \(P_{1}=(u_{1},\,u_{2},\,...,\,u_{n(G)})\) and \(P_{2}=(v_{1},\,v_{2},\,...,\,v_{n(G)})\) of \(G\) from \(u\) to \(v\) are independent if \(u=u_{1}=v_{1}\), \(v=u_{n(G)}=v_{n(G)}\), and \(u_{i}\
eq v_{i}\), for every \(1<i<n(G)\). A set of Hamiltonian paths, \(\{P_{1},\,P_{2},\,...,\,P_{k}\}\), of \(G\) from \(u\) to \(v\) is _mutually independent_ if any two different _Hamiltonian paths_ are independent from \(u\) to \(v\). The mechanism of mutually independent Hamiltonian paths (MIHP) can be applied to parallel processing [18]. Such a feature is also considered for secret communications [19,20].