A well-equalized 3-CIST partition of alternating group graphs

https://doi.org/10.1016/j.ipl.2019.105874Get rights and content

Highlights

  • The problem of constructing completely independent spanning trees on alternating group graph AGn was investigated.

  • We disclose a property called well-equalized 3-CIST partition of AGn.

  • As a by-product, we obtain a constructing scheme of three CISTs of AGn for n5.

Abstract

A set of k (2) spanning trees in a graph is called completely independent spanning trees (CISTs for short) if they are pairwise edge-disjoint and internally vertex-disjoint. Although the problem of determining whether a graph G admits k CISTs is NP-complete even for k=2, there exists a characterization called k-CIST partition for which the instances can easily be verified. A k-CIST partition of a graph G=(V,E) is a partition of V into V1,V2,,Vk such that the subgraph of G induced by Vi, denoted by G[Vi], is connected for i{1,2,,k}, and the bipartite graph induced by the edge set {(x,y)E:xVi,yVj}, denoted by B(Vi,Vj,G), has no tree component for i,j{1,2,,k} with ij. Moreover, a k-CIST partition is well-equalized provided all G[Vi] are isomorphic for i{1,2,,k} and all B(Vi,Vj,G) are isomorphic for distinct i,j{1,2,,k}. The class of alternating group graphs is a kind of well-studied interconnection networks, which forms a famous subclass of Cayley graphs and possesses a large number of desirable properties of networks. Let AGn denote the n-dimensional alternating group graph. In this paper, we show that AGn admits a well-equalized 3-CIST partition. As a by-product, we obtain a constructing scheme of three CISTs for AGn.

Introduction

Let k2 be an integer and T1,T2,,Tk be spanning trees of a graph G=(V,E). A vertex in a tree Ti is a leaf if it has degree one, and an internal vertex otherwise. Two spanning trees Ti and Tj are edge-disjoint if they share no common edge and internally vertex-disjoint provided the paths joining any two vertices u,vV in both trees have no common vertex except for u and v. The spanning trees T1,T2,,Tk are completely independent spanning trees (CISTs for short) if they are pairwise edge-disjoint and internally vertex-disjoint.

Motivated by applications in fault-tolerant communication, the study of CISTs was due to an early work of Hasunuma [10], [11]. In [11], Hasunuma pointed out that determining if a graph G admits k CISTs is an NP-complete problem even for k=2. Hence, finding sufficient conditions for graphs that admit multiple CISTs were investigated in [1], [2], [8], [12], [14]. Especially, the degree-based sufficient conditions such as Dirac's condition [1] and Ore's condition [8]. Péterfalvi [21] showed that for any k2, there exists a k-connected graph which does not possess two CISTs. This disproved the conjecture posed by Hasunuma [11], which states that there exist k CISTs in a 2k-connected graph (see also [20] for more counterexamples). Conversely, with the help of constructions, it has been confirmed that certain classes of graphs possess two CISTs, e.g., 4-connected maximal planar graphs [11], Cartesian product of any 2-connected graphs [13], hypercube-variant networks [6], [18], 4-regular chordal rings [3], [20]. In addition, more graphs possessing multiple CISTs can be found in [7], [10], [17], [19].

The following two characterizations are important for studying CISTs.

Theorem 1

(Hasunuma [10]) For a graph G=(V,E), a set of spanning trees T1,T2,,Tk are CISTs of G if and only if they are edge-disjoint and for any vertex vV, there is at most one spanning tree Ti such that v is an internal vertex of Ti.

Theorem 2

(Araki [1]) A graph G=(V,E) admits k CISTs if and only if there is a partition of V into V1,V2,,Vk, which is called the k-CIST partition, such that the following conditions hold:

  • (i)

    For i{1,2,,k}, the subgraph of G induced by Vi, denoted by G[Vi], is connected;

  • (ii)

    For distinct i,j{1,2,,k}, the bipartite graph with bipartition ViVj and edge set {(x,y)E(G):xVi,yVj}, denoted by B(Vi,Vj,G), has no tree component.

For a graph G=(V,E), a k-CIST partition is called an equalized partition if |Vi|=|V|/k for all 1ik. Moreover, an equalized k-CIST partition is well-equalized provided all G[Vi] are isomorphic for i{1,2,,k} and all bipartite graphs B(Vi,Vj,G) are isomorphic for i,j{1,2,,k} with ij. In this paper, we show that the n-dimensional alternating group graph for n5 admits the property of well-equalized 3-CIST partition.

Section snippets

Alternating group graphs

For n3, let Zn={1,2,,n} and p=p1p2pn be a permutation of elements of Zn, where piZn is the symbol at the position i in the permutation. Two distinct symbols pi and pj are said to be a pair of inversion of p if (pipj)(ij)<0. A permutation is an even (resp. odd) permutation provided it has an even (resp. odd) number of inversions. We denote An the set of all even permutations over Zn. Let gij denote the transposition of a permutation that swaps symbols at positions i and j and leaves all

A well-equalized 3-CIST partition of AG5

In this section, we show that AG5 admits a well-equalized 3-CIST partition. For high-dimensional AGn with n6, all 3-CIST partitions will be constructed by a recursive fashion (see Section 4). Hence, the result presented here can be viewed as the induction base.

Recall that An denotes the set of all even permutations over Zn. Clearly, |A5|=60. To find an equalized 3-CIST-partition of AG5, we are looking forward to obtaining a partition such that |V1|=|V2|=|V3|=|A5|/3=20. Let x0=12345, y0=12453

A well-equalized 3-CIST partition on high-dimensional AGn

In this section, we consider n6 and provide a recursive construction for building a well-equalized 3-CIST partition of AGn. Let p=p1p2pn1An1. We denote by p{n} the permutation obtained from p by adding a suffix symbol n. It is clear that p{n}An. Recall that AGnk is the subgraph of AGn induced by vertices with rightmost symbol k. We say that a subgraph AGnk is hereditary provided k=n, and non-hereditary otherwise. For p=p1p2pn1V(AGn1) and kZn, the agent of p in AGnk, denoted α(p,k),

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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    This research was partially supported by MOST grants 107-2221-E-131-011 (K.-J. Pai), 107-2221-E-141-002 (R.-S. Chang) and 107-2221-E-141-001-MY3 (J.-M. Chang), from the Ministry of Science and Technology, Taiwan.

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