Affiliation:
1. Department of Mathematics, Savitribai Phule Pune University, Pune 411007, India
Abstract
We consider the problem of determining the possible orders for [Formula: see text]-regular, [Formula: see text]-connected and bipancyclic subgraphs of the hypercube [Formula: see text] For [Formula: see text] and [Formula: see text] the solution to the problem is known. In this paper, we solve the problem for [Formula: see text] by proving that [Formula: see text] has a 4-regular, 4-connected and bipancyclic subgraph on [Formula: see text] vertices if and only if [Formula: see text] or [Formula: see text] is an even integer such that [Formula: see text] Further, by improving a result of Ramras, we prove that a [Formula: see text]-regular subgraph of [Formula: see text] is either isomorphic to [Formula: see text] or has at least [Formula: see text] vertices. We also improve a result of Mane and Waphare regarding the existence of a [Formula: see text]-regular, [Formula: see text]-connected and bipancyclic subgraph of [Formula: see text] Some applications of our results are given.
Funder
Department of Science and Technology, SERB, Government of India
Publisher
World Scientific Pub Co Pte Lt
Subject
Discrete Mathematics and Combinatorics
Cited by
7 articles.
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1. A Note on Conditional Edge Connectivity of Hypercube Networks;Journal of Interconnection Networks;2021-06
2. On Edge-Fault Tolerance in Augmented Cubes;Journal of Interconnection Networks;2020-12
3. On regular subgraphs of augmented cubes;AKCE International Journal of Graphs and Combinatorics;2020-04-27
4. On conditional connectivity of the Cartesian product of cycles;Discussiones Mathematicae Graph Theory;2020
5. Existence of 3-regular subgraphs in Cartesian product of cycles;AKCE International Journal of Graphs and Combinatorics;2019-12-01