A well-equalized 3-CIST partition of alternating group graphs☆
Introduction
Let be an integer and be spanning trees of a graph . A vertex in a tree is a leaf if it has degree one, and an internal vertex otherwise. Two spanning trees and are edge-disjoint if they share no common edge and internally vertex-disjoint provided the paths joining any two vertices in both trees have no common vertex except for u and v. The spanning trees 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 . 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 , 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 , a set of spanning trees are CISTs of G if and only if they are edge-disjoint and for any vertex , there is at most one spanning tree such that v is an internal vertex of .
Theorem 2 (Araki [1]) A graph admits k CISTs if and only if there is a partition of V into , which is called the k-CIST partition, such that the following conditions hold: For , the subgraph of G induced by , denoted by , is connected; For distinct , the bipartite graph with bipartition and edge set , denoted by , has no tree component.
For a graph , a k-CIST partition is called an equalized partition if for all . Moreover, an equalized k-CIST partition is well-equalized provided all are isomorphic for and all bipartite graphs are isomorphic for with . In this paper, we show that the n-dimensional alternating group graph for admits the property of well-equalized 3-CIST partition.
Section snippets
Alternating group graphs
For , let and be a permutation of elements of , where is the symbol at the position i in the permutation. Two distinct symbols and are said to be a pair of inversion of p if . A permutation is an even (resp. odd) permutation provided it has an even (resp. odd) number of inversions. We denote the set of all even permutations over . Let denote the transposition of a permutation that swaps symbols at positions i and j and leaves all
A well-equalized 3-CIST partition of
In this section, we show that admits a well-equalized 3-CIST partition. For high-dimensional with , 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 denotes the set of all even permutations over . Clearly, . To find an equalized 3-CIST-partition of , we are looking forward to obtaining a partition such that . Let ,
A well-equalized 3-CIST partition on high-dimensional
In this section, we consider and provide a recursive construction for building a well-equalized 3-CIST partition of . Let . We denote by the permutation obtained from p by adding a suffix symbol n. It is clear that . Recall that is the subgraph of induced by vertices with rightmost symbol k. We say that a subgraph is hereditary provided , and non-hereditary otherwise. For and , the agent of p in , denoted ,
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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2022, Applied Mathematics and ComputationCitation Excerpt :It possesses such attractive properties like node symmetry, edge symmetry, and small diameter [5,11]. Many scholars do research on various properties of alternating group graph recent years, including the pancyclicity and hamiltonian-connectivity [5], the fault-tolerant pancyclicity, node-pancyclicity and edge-pancyclicity [1,2,12–14], the hamiltonian cycle properties [15], the disjoint path covers [19], the generalized connectivity [21], the conditional diagnosability [3], the extra diagnosability [4], the good-neighbor diagnosability [4], the 2-restricted connectivity [20], the well-equalized 3-CIST (completely independent spanning trees) [10], the structure and substructure connectivity [8], the 2-extra diagnosability [17], the automorphism group [22], and the pessimistic diagnosability [16]. The paper contains four sections.
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2020, Journal of Parallel and Distributed ComputingCitation Excerpt :A graph with a dual-CIST is protectable. From Theorem 3 and the known results of existing dual-CISTs in literature, we directly obtain the corollary that the following classes of graphs (or networks) are protectable: the underlying graph of a 2-connected line digraph [13], 4-connected maximal planar graph [14], the Cartesian product of any 2-connected graphs [16], complete graphs, complete bipartite graphs and complete tripartite graphs with connectivity at least 4 [32], the square of 2-connected graphs [1], the 4-dimensional hypercube-variant networks (including hypercubes, locally twisted cubes, crossed cubes, parity cubes, and Möbius cubes) [28], augmented cubes [25], balanced hypercubes [46], Cayley graphs (including star graphs, bubble-sort graphs, alternating group networks [29], alternating group graphs [31]), and data center networks DCell [36] and HSDC [37]. Note that another application of protection routing that uses a combination of multiple CISTs to increase the capability of fault-tolerance will be demonstrated in Section 5.
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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.