Mutually independent bipanconnected property of hypercube

Abstract

A graph is denoted by G with the vertex set V(G) and the edge set E(G). A path P = 〈v₀, v₁, … , v_m〉 is a sequence of adjacent vertices. Two paths with equal length P₁ = 〈 u₁, u₂, … , u_m〉 and P₂ = 〈 v₁, v₂, … , v_m〉 from a to b are independent if u₁ = v₁ = a, u_m = v_m = b, and u_i ≠ v_i for 2 ⩽ i ⩽ m − 1. Paths with equal length $\{P_i{\}}_{i=1}^n$ from a to b are mutually independent if they are pairwisely independent. Let u and v be two distinct vertices of a bipartite graph G, and let l be a positive integer length, d_G(u, v) ⩽ l ⩽ ∣V(G) − 1∣ with (l − d_G(u, v)) being even. We say that the pair of vertices u, v is (m, l)-mutually independent bipanconnected if there exist m mutually independent paths ${\left\{P_i^l\right\}}_{i=1}^m$ with length l from u to v. In this paper, we explore yet another strong property of the hypercubes. We prove that every pair of vertices u and v in the n-dimensional hypercube, with $d_{Q_n}(u\text{,}v)⩾n-1$, is (n − 1, l)-mutually independent bipanconnected for every $l\text{,}{d}_{Q_n}(u\text{,}v)⩽l⩽|V(Q_n)-1|$ with $(l-d_{Q_n}(u\text{,}v))$ being even. As for $d_{Q_n}(u\text{,}v)⩽n-2$, it is also (n − 1, l)-mutually independent bipanconnected if $l⩾d_{Q_n}(u\text{,}v)+2$, and is only (l, l)-mutually independent bipanconnected if $l=d_{Q_n}(u\text{,}v)$.

Introduction

For the graph definitions and notations we refer the reader to [10]. A graph is denoted by G with the vertex set V(G) and the edge set E(G). The simulation of one architecture by another is an important issue in interconnection networks. The problem of simulating one network by another is also called embedding problem. One particular problem of path embedding deals with finding all the possible length of paths in an interconnection network.

A path P = 〈v₀, v₁, … , v_m〉 is a sequence of adjacent vertices. We also write P = 〈v₀, … , v_i, Q, v_j, … , v_m 〉 where Q is a path 〈v_i, … , v_j〉. A cycle C = 〈v₀, v₁, … , v_m, v₀〉 is a sequence of adjacent vertices where the first vertex is the same as the last one. The length of a path P (a cycle C respectively) is the number of edges in P (in C respectively).

A cycle of G is a hamiltonian cycle if it traverses all the vertices exactly once. A graph G is called a hamiltonian graph if G contains a hamiltonian cycle. There are many studies about the hamiltonian graphs [3], [4], [15]. A path of G is a hamitonian path if it contains all the vertices exactly once. A graph G is hamiltonian connected if there exists a hamiltonian path between any two different vertices of G. A graph G = (B ∪ W, E) is bipartite if V(G) is the union of two disjoint sets B and W such that every edge joins B with W. It is easy to see that any bipartite graph with at least three vertices is not hamiltonian connected. A bipartite graph G is hamiltonian laceable if there exists a hamiltonian path joining any two vertices from different partite sets. A graph G is pancyclic [2] if G includes cycles of all lengths. If these cycles are restricted to even length, G is called a bipancyclic graph. The distance from x to y, written d_G(x, y), is the least length among all paths from x to y in G. A graph is panconnected if, for any two different vertices x and y, there exists a path of length l joining x and y, for every l, d_G(x, y) ⩽ l ⩽ ∣V(G)∣ − 1. The concept of panconnected graphs is proposed by Alavi and Williamson [1]. Recently, there are many studies about pancyclicity and panconnectivity of graphs [5], [6].

It is not hard to see that any bipartite graph with at least 3 vertices is not panconnected. Therefore, the concept of bipanconnected graphs is proposed. A bipartite graph is bipanconnected if, for any two different vertices x and y, there exists a path of length l joining from x to y, for every l, d_G(x, y) ⩽ l ⩽  ∣V(G)∣ − 1 and (l − d_G(x, y)) being even. There are many studies on bipanconnected graphs and bipancyclic graphs [7], [11], [13], [18].

We introduce some terms defined recently. Two paths P₁ = 〈u₁, u₂, … , u_m〉 and P₂ = 〈v₁, v₂, … , v_m〉 from a to b are independent [14] if u₁ = v₁ = a, u_m = v_m = b, and u_i ≠ v_i for 2 ⩽  i ⩽ m − 1. Paths with equal length $\{P_i{\}}_{i=1}^n$ from a to b are mutually independent [14] if they are pairwisely independent. Two cycles C₁ = 〈u₁, u₂, … , u_m, u₁〉 and C₂ = 〈v₁, v₂, … , v_m, v₁〉 beginning at x are independent if u₁ = v₁ = x and u_i ≠ v_i for 2 ⩽ i  ⩽ m. Cycles with equal length $\{C_i{\}}_{i=1}^n$ beginning at x are mutually independent if every two cycles are independent. Two hamiltonian paths P₁ = 〈 u₁, u₂, … , u_∣V(G)∣〉 and P₂ = 〈 v₁, v₂, … , v_∣V(G)∣〉 are independent beginning at x [9] if u₁ = v₁ = x and u_i ≠ v_i for 2 ⩽ i ⩽ ∣V(G)∣, denoted P₁: x → u_∣V(G)∣ and P₂: x → v_∣V(G)∣. Hamiltonian paths $\{P_i{\}}_{i=1}^n$ are mutually independent hamiltonian paths beginning at x [9] if any two of them are independent beginning at x.

An n-dimensional hypercube, denoted by Q_n, is a graph with 2ⁿ vertices, and each vertex u can be distinctly labeled by an n-bit binary string, u = u_n−1u_n−2, … , u₁u₀. There is an edge between two vertices if and only if their binary labels differ in exactly one bit position. Let (u, v) be an edge in Q_n. If the binary labels of u and v differ in ith position, then the edge between them is said to be in ith dimension and the edge (u, v) is called an ith dimension edge. We use $Q_{n-1}^0$ to denote the subgraph of Q_n induce by {u ∈ V(Q_n)∣u_i = 0} and $Q_{n-1}^1$ to denote the subgraph of Q_n induced by $\{u\in V(Q_n)|u_i=1\}.Q_{n-1}^0$ and $Q_{n-1}^1$ are all isomorphic to Q_n−1.Q_n can be decomposed into $Q_{n-1}^0$ and $Q_{n-1}^1$ by dimension i, and $Q_{n-1}^0$ and $Q_{n-1}^1$ are (n − 1)-dimensional subcubes of Q_n induced by the vertices with the ith bit position being 0 and 1 respectively. For each vertex u in $Q_{n-1}^i\text{,}i=\{0\text{,}1\}$, there is exactly one vertex in $Q_{n-1}^{|i-1|}$, denoted by $\overline{u}$, such that $(u\text{,}\overline{u})$ is an edge in Q_n. There are many studies on the hypercubes [9], [13], [16], [17], [19], [20].

We now introduce a new concept. Let u and v be two distinct vertices of a bipartite graph G and let l be a positive integer length, d_G(u, v) ⩽ l ⩽ ∣V(G) − 1∣ with (l − d_G(u, v)) being even. We say that the pair of vertices u, v is (m, l)-mutually independent bipanconnected if there exist m mutually independent paths ${\left\{P_i^l\right\}}_{i=1}^m$ with length l from u to v. In this paper,we explore yet another strong property of the hypercubes. We prove that every pair of vertices u and v in the n-dimensional hypercube, with $d_{Q_n}(u\text{,}v)⩾n-1$, is (n − 1, l)-mutually independent bipanconnected for every l, $d_{Q_n}(u\text{,}v)⩽l⩽|V(Q_n)-1|$ with $(l-d_{Q_n}(u\text{,}v))$ being even. As for $d_{Q_n}(u\text{,}v)⩽n-2$, it is also (n − 1, l)-mutually independent bipanconnected if $l⩾d_{Q_n}(u\text{,}v)+2$, and is only (l, l)-mutually independent bipanconnected if $l=d_{Q_n}(u\text{,}v)$. Our result strengthens a previous results of Sun et al. [19], and Li et al. [13]. Li et al. [13] proved that the hypercube Q_n is bipanconnected for n ⩾ 2. Sun et al. [19] proved that there are n − 1 mutually independent hamiltonian paths in Q_n between every two vertices from different partite sets for n  ⩾ 4. The number “n − 1” in our result is tight as we have the following observation. Because each vertex of the hypercube Q_n has exactly n edges incident with it, we can expect at most n − 1 mutually independent paths when the given two vertices are adjacent.