International Journal of Combinatorics

Volume 2015, Article ID 638767, 4 pages

http://dx.doi.org/10.1155/2015/638767

## On 3-Regular Bipancyclic Subgraphs of Hypercubes

Department of Mathematics, University of Pune, Pune 411 007, India

Received 31 July 2014; Accepted 15 April 2015

Academic Editor: Chris A. Rodger

Copyright © 2015 Y. M. Borse and S. R. Shaikh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The -dimensional hypercube is *bipancyclic*; that is, it contains a cycle of every even length from 4 to . In this paper, we prove that contains a 3-regular, 3-connected, bipancyclic subgraph with vertices for every even from 8 to except 10.

#### 1. Introduction

The* cartesian product * of two graphs and is a graph with the vertex set , and any two vertices and are adjacent in if and only if either and is adjacent to in or and is adjacent to in . A graph with even number of vertices is* bipancyclic* if it contains a cycle of every even length from 4 to . The* hypercube * of dimension is a graph obtained by taking cartesian product of the complete graph on two vertices with itself times; that is, ( times). The hypercube is an -regular, -connected, bipartite, and bipancyclic graph with vertices. It is one of the most popular interconnection network topologies [1]. The bipancyclicity of a given network is an important factor in determining whether the network topology can simulate rings of various lengths. The connectivity of a network gives the minimum cost to disrupt the network. Regular subgraphs, bipancyclicity, and connectivity properties of hypercubes are well studied in the literature [2–6].

Since () is bipancyclic, it contains a 2-regular, 2-connected subgraph (cycle) with vertices for every even integer from 4 to . Suppose . Mane and Waphare [4] proved that contains a spanning -regular, -connected, bipancyclic subgraph. So the natural question arises; what are the other possible orders existing for -regular, -connected and bipancyclic subgraphs of As , can be regarded as a subgraph of . Hence has a -regular, -connected, bipancyclic subgraph with vertices. In this paper, we answer the question for . We prove that () contains a -regular, -connected, and bipancyclic subgraph with vertices for every even integer from 8 to except 10.

#### 2. Proof

The cartesian product of a nontrivial path with the complete graph is a* ladder* graph. Let be the graph obtained from a path () by adding one extra edge . We call the graph a* ladder type * graph on vertices (see Figure 1).