Two disjoint cycles of various lengths in alternating group graph

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Highlights

  • Two-disjoint-cycle-cover pancyclicity is generalization of pancyclicity.

  • The n-dimensional alternating group graph AGn is an important network topology.

  • The AGn is two-disjoint-cycle-cover [3;n!4]-pancyclicity, where n4.

Abstract

Alternating group graph has been widely studied recent years because it possesses many good properties. For a graph G, the two-disjoint-cycle-cover [r1,r2]-pancyclicity refers that it contains cycles C1 and C2, where V(C1)V(C2)=, (C1)+(C2)=|V(G)| and r1(C1)r2. In this paper, it is proved that the n-dimensional alternating group graph AGn is two-disjoint-cycle-cover [3,n!4]-pancyclic, where n4.

Introduction

A simple graph is a way to express the interconnection network, where the processors and links are represented by the vertices and edges respectively. The alternating group graph [5] as an important interconnection network topology has attracted many researchers’ interest. 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], [13], [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].

For a graph G, the two-disjoint-cycle-cover [r1,r2]pancyclicity refers that it contains two cycles C1 and C2, where V(C1)V(C2)=, (C1)+(C2)=|V(G)|, and r1(C1)r2 [6], [7]. We suppose that (C1)(C2), then r2|V(G)|2. In bipartite graph, the two-disjoint-cycle-cover [r1,r2]-bipancyclicity can be defined similarly [18]. The two-disjoint-cycle-cover vertex [r1,r2]-(bi)pancyclicity of (bipartite) graph G refers that for any two different vertices, it contains two vertex disjoint (even) cycles C1 and C2 which includes one of these two vertices respectively, where |V(C1)|+|V(C2)|=|V(G)| and r1(C1)r2 [9]. The above definitions are considered as generalizations to pancyclicity, bipancyclicity, vertex pancyclicity and vertex bipancyclicity respectively. The related properties are studied on crossed cube [7], locally twisted cube [6], balanced hypercubes [18] and the bipartite generalized hypercube [9] recently. Inspired by the previous results, we determine the two-disjoint-cycle-cover pancyclicity of alternating group graph.

The paper contains four sections. In next section, terminologies needed are presented. The main results and conclusion are shown in Sections 3 and 4, respectively.

Section snippets

Perliminaries

A graph G contains vertex set V(G) and edge set E(G). The path from vertex u0 to un is written as P[u0,un]=u0,u1,u2,,un1,un, in which uiuj for ij and (uk,uk+1)E(G), where 0kn1. Furthermore, if u0=un and n2, then P[u0,un] becomes a cycle. The number of edges on path P (respectively, cycle C) is its length, and is written as (P) (respectively, (C)). The distance from vertex u to v, is d(u,v)=min{(P[u,v])}. A graph G is called panconnected if for any two vertices, it contains paths

Main results

We firstly prove the following lemmas.

Lemma 7

Let AGn=AGn11AGn12AGn1n. For any internal edge in E(AGn), its two end vertices respectively have one external neighbour that are adjacent in the same (n1)-dimensional sub-alternating group graph, and respectively have one external neighbour in the other two different (n1)-dimensional sub-alternating group graphs.

Proof

By Lemma 1, AGn is edge symmetric. Without loss of generality, let e=(s,t)E(AGn1x), where s=zywx and t=ywzx. Then by Lemma 3, s=sgn+

Conclusion

It is obtained that the AGn is two-disjoint-cycle-cover [3,n!4]-pancyclic when n4. Because |V(AGn)|=n!2, this result is optimal.

Acknowledgments

The author expresses her gratitude to the anonymous referees for their valuable suggestions which greatly improve the final version of this paper. This work was supported by China Scholarship Council (CSC) under Grant ( 202006785015).

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