Two disjoint cycles of various lengths in alternating group graph
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 the two-disjoint-cycle-cover pancyclicity refers that it contains two cycles and where and [6], [7]. We suppose that then . In bipartite graph, the two-disjoint-cycle-cover -bipancyclicity can be defined similarly [18]. The two-disjoint-cycle-cover vertex -(bi)pancyclicity of (bipartite) graph refers that for any two different vertices, it contains two vertex disjoint (even) cycles and which includes one of these two vertices respectively, where and [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 contains vertex set and edge set . The path from vertex to is written as in which for and where . Furthermore, if and then becomes a cycle. The number of edges on path (respectively, cycle ) is its length, and is written as (respectively, ). The distance from vertex to is min. A graph is called panconnected if for any two vertices, it contains paths
Main results
We firstly prove the following lemmas. Lemma 7 Let . For any internal edge in its two end vertices respectively have one external neighbour that are adjacent in the same -dimensional sub-alternating group graph, and respectively have one external neighbour in the other two different -dimensional sub-alternating group graphs. Proof By Lemma 1, is edge symmetric. Without loss of generality, let where and . Then by Lemma 3,
Conclusion
It is obtained that the is two-disjoint-cycle-cover -pancyclic when . Because 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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