On mutually independent hamiltonian paths

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www.elsevier.com/locate/aml

On mutually independent hamiltonian paths

Yuan-Hsiang Teng

a

, Jimmy J.M. Tan

a,∗

, Tung-Yang Ho

b

, Lih-Hsing Hsu

c a_{Department of Computer and Information Science, National Chiao Tung University, Hsinchu City 300, Taiwan, ROC} b_{Department of Industrial Engineering and Management, Ta Hwa Institute of Technology, Hsinchu County 307, Taiwan, ROC}

c_{Department of Computer Science and Information Engineering, Ta Hwa Institute of Technology, Hsinchu County 307,} Taiwan, ROC

Received 14 April 2005; accepted 12 May 2005

Abstract

Let P1 = v1, v2, v3, . . . , vn and P2 = u1, u2, u3, . . . , un be two hamiltonian paths of G. We say that P1

and P2 are independent if u1 = v1, un = vn, and ui = vi for 1 < i < n. We say a set of hamiltonian paths

P1, P2, . . . , Ps of G between two distinct vertices are mutually independent if any two distinct paths in the set

are independent. We use n to denote the number of vertices and use e to denote the number of edges in graph G. Moreover, we use ¯e to denote the number of edges in the complement of G. Suppose that G is a graph with ¯e ≤ n − 4 and n ≥ 4. We prove that there are at least n − 2 − ¯e mutually independent hamiltonian paths between

any pair of distinct vertices of G except n= 5 and ¯e = 1. Assume that G is a graph with the degree sum of any two non-adjacent vertices being at least n+ 2. Let u and v be any two distinct vertices of G. We prove that there are deg_G(u) + deg_G(v) − n mutually independent hamiltonian paths between u and v if (u, v) ∈ E(G) and there are deg_G(u) + deg_G(v) − n + 2 mutually independent hamiltonian paths between u and v if otherwise.

Keywords: Hamiltonian; Hamiltonian connected; Hamiltonian path

1. Definitions and notation

For the graph definition and notation we follow [1]. G = (V, E) is a graph if V is a finite set and

E is a subset of{(u, v) | (u, v) is an unordered pair of V }. We say that V is the vertex set and E is the ∗_{Corresponding author.}

E-mail address: [email protected] (J.J.M. Tan).

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aspect of hamiltonian connected graphs. Let P1 = v1, v2, v3, . . . , vn and P2 = u1, u2, u3, . . . , un

be any two hamiltonian paths of G. We say that P1 and P2 are independent if u1 = v1, un = vn, and ui = vi for 1< i < n. We say a set of hamiltonian paths P1, P2, . . . , Psof G are mutually independent

if any two distinct paths in the set are independent. In [4], it is proved that there exist(k − 2) mutually independent hamiltonian paths between any two vertices from different bipartite sets of the star graph

Sk if k ≥ 4. The concept of mutually independent hamiltonian arises from the following application. If

there are k pieces of data needed to be sent from u tov, and the data needed to be processed at every node (and the process takes times), then we want mutually independent hamiltonian paths so that there will be no waiting time at a processor. The existence of mutually independent hamiltonian paths is useful for communication algorithms. Motivated by this result, we begin the study on graphs with mutually independent hamiltonian paths between every pair of distinct vertices.

In this work, we are interested in two families of graphs. The first family of graphs ¯e ≤ n − 4. It was proved [5] that such graphs are hamiltonian connected. In this work, we strengthen this classical result by proving that there are at least n−2− ¯e mutually independent hamiltonian paths between every pair of distinct vertices of G. The second family of graphs are those graphs with the sum of the degree of any two non-adjacent vertices being at least n+ 1. It was proved [3] that such graphs are hamiltonian connected. We then further assume that G is a graph with the sum of any two non-adjacent vertices being at least

n+ 2. Let u and v be any two distinct vertices of G. Then there are degG(u) + degG(v) − n mutually

independent hamiltonian paths between u andv if (u, v) ∈ E(G), and there are degG(u)+degG(v)−n+2 mutually independent hamiltonian paths between u andv otherwise.

Throughout this work, we will use[i] to denote i mod (n − 2).

2. Preliminary

Let G and H be two graphs. We use G + H to denote the disjoint union of G and H. We use

G∨ H to denote the graph obtained from G + H by joining each vertex of G to each vertex of H. For

1 ≤ m < n/2, let Cm,n denote the graph( ¯Km + Kn−2m) ∨ Km; seeFig. 1. The following theorem is

proved by Chvátal [2].

Theorem 1 ([2]). Assume that G is a graph with n≥ 3 and ¯e ≤ n−3. Then G is hamiltonian. Moreover, the only non-hamiltonian graphs with ¯e ≤ n − 2 are C1,n and C2,5.

The following lemma is obvious.

Lemma 1. Let u and v be two distinct vertices of G. Then there are at most min{degG(u), degG(v)} mutually independent hamiltonian paths between u and v if (u, v) ∈ E(G), and there are at most

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Fig. 1. Cm,n.

Theorem 2. Let n be a positive integer with n ≥ 3. There are n − 2 mutually independent hamiltonian paths between every two distinct vertices of Kn.

Proof. Let s and t be two distinct vertices of Kn. We relabel the remaining(n − 2) vertices of Kn as

0, 1, 2, . . . , n − 3. For 0 ≤ i ≤ n − 3, we set Pi ass, [i], [i + 1], [i + 2], . . . , [i + (n − 3)], t. It is easy to see that P0, P1, . . . , Pn−3form(n − 2) mutually independent hamiltonian paths joining s and t. Theorem 3 ([5]). Assume that G is a graph with¯e ≤ n−4 and n ≥ 4. Then G is hamiltonian connected. Theorem 4 ([5]). Assume that G is a graph with the sum of any two distinct non-adjacent vertices being at least n with n≥ 3. Then G is hamiltonian.

Theorem 5 ([3]). Assume that G is a graph with the sum of any two distinct non-adjacent vertices being at least n+ 1 with n ≥ 3. Then G is hamiltonian connected.

3. Mutually independent hamiltonian paths

The following result strengthens that ofTheorem 3.

Lemma 2. Assume that G is a graph with n ≥ 4 and ¯e = n − 4. Then there are two independent hamiltonian paths between any two distinct vertices of G except n= 5.

Proof. For n = 4, G is isomorphic to K4. ByTheorem 2, there are two independent hamiltonian paths

between any two distinct vertices of G. Assume that n= 5. Then G is isomorphic to K5− { f } for some

edge f . Without loss of generality, we assume that V(G) = {1, 2, 3, 4, 5} and f = (1, 2). It is easy to check that P1 = 3, 2, 5, 1, 4 and P2= 3, 1, 5, 2, 4 are the only two hamiltonian paths between 3 and

4, but P1and P2are not independent.

Now, we assume that n ≥ 6. Let s and t be any two distinct vertices of G. Let H be the subgraph of

G induced by the remaining(n − 2) vertices of G. We have the following two cases:

Case 1: H is hamiltonian. We can relabel the vertices of H with {0, 1, 2, . . . , n − 3} so that 0, 1, 2, . . . , n − 3, 0 forms a hamiltonian cycle of H. Let Q denote the set {i | (s, [i + 1]) ∈ E(G) and (i, t) ∈ E(G)}. Since ¯e = n −4, |Q| ≥ n −2−(n −4) = 2. There are at least two elements in Q. Let q1and q2be the two elements in Q. For j = 1, 2, we set Pj ass, [qj+ 1], [qj+ 2], . . . , [qj], t.

Then P1and P2are two independent hamiltonian paths between s and t.

Case 2: H is non-hamiltonian. There are exactly(n − 2) vertices in H. ByTheorem 1, there are exactly

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Fig. 2. (a) C₂_,5, (b) C₁_,n−2.

know that(s, v) ∈ E(G) and (t, v) ∈ E(G) for every vertex v in H. We can construct two independent hamiltonian paths between s and t as following cases:

Subcase 2.1: H is isomorphic to C2,5. We label the vertices of C2,5 with {0, 1, 2, 3, 4} as shown in

Fig. 2(a). Let P1 = s, 0, 1, 2, 3, 4, t and P2 = s, 2, 3, 4, 1, 0, t. Then P1 and P2 form the required

independent paths.

Subcase 2.2: H is isomorphic to C1,n−2. We label the vertices of C1,n−2with{0, 1, . . . , n −3} as shown

inFig. 2(b). Let P1 = s, 0, 1, 2, . . . , n − 3, t and P2 = s, 2, 3, . . . , n − 3, 1, 0, t. Then P1 and P2

form the required independent paths. We can further strengthenTheorem 3:

Theorem 6. Assume that G is a graph with n ≥ 4 and ¯e ≤ n − 4. Then there are n − 2 − ¯e mutually independent hamiltonian paths between every two distinct vertices of G except n = 5 and ¯e = 1. Proof. With Lemma 2, the theorem for ¯e = n − 4 holds. Now, we need to prove the theorem for

¯e = n − 4 − r with 1 ≤ r ≤ n − 4. Let s and t be two distinct vertices of G. Let H be the subgraph of G induced by the remaining(n − 2) vertices of G.

Then there are exactly(n − 2) vertices in H and there are at most n − 4 − r edges in the complement of H with 1 ≤ r ≤ n − 4. By Theorem 1, H is hamiltonian. We can label the vertices of H with

{0, 1, 2, . . . , n − 3} so that 0, 1, 2, . . . , n − 3, 0 forms a hamiltonian cycle of H. Let Q denote the

set {i | (s, [i + 1]) ∈ E(G) and (t, i) ∈ E(G)}. Since ¯e = n − 4 − r with 1 ≤ r ≤ n − 4, we know that |Q| ≥ n − 2 − (n − 4 − r) = n − 2 − ¯e for 1 ≤ r ≤ n − 4. Hence, there are at least

n− 2 − ¯e elements in Q. Let q1, q2, . . . , qn−2−¯e be the elements in Q. For j = 1, 2, . . . , n − 2 − ¯e, we

set Pj = s, [qj+1], [qj+2], . . . , [qj], t. It is not difficult to see that P1, P2, . . . , Pn−2−¯eare mutually

independent paths between s and t.

The following result, in a sense, generalizes that ofTheorem 5.

Theorem 7. Assume that G is a graph such that degG(x) + degG(y) ≥ n + 2 for any two vertices x and y with(x, y) ∈ E(G). Let u and v be two distinct vertices of G. Then there are degG(u) + degG(v) − n

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mutually independent hamiltonian paths between u andv if (u, v) ∈ E(G), and there are degG(u) +

degG(v) − n + 2 mutually independent hamiltonian paths between u and v if (u, v) ∈ E(G).

Proof. Let s and t be two distinct vertices of G, and H be the subgraph of G induced by the remaining (n − 2) vertices of G. Let u _and_v _{be any two distinct vertices in H . We have deg}

H(u ) + degH(v ) ≥ n+ 2 − 4 = n − 2 = |V (H)|. ByTheorem 4, H is hamiltonian. We can label the vertices of H with

{0, 1, . . . , n − 3}, so that 0, 1, 2, . . . , n − 3, 0 forms a hamiltonian cycle of H. Let S denote the set {i | (s, [i + 1]) ∈ E(G)} and T denote the set {i | (i, t) ∈ E(G)}. Clearly, |S ∪ T | ≤ n − 2. We have the

following two cases:

Case 1:(s, t) ∈ E(G). Suppose that |S ∩T | ≤ degG(s)+degG(t)−n −1. We have degG(s)+degG(t)−

2= |S| + |T | = |S ∪ T | + |S ∩ T | ≤ degG(s) + degG(t) − n − 1 + n − 2. This is a contradiction. Thus, there are at leastw = deg_G(s) + deg_G(t) − n elements in S ∩ T . Let q1, q2, . . . , qw be the elements

in S∩ T . For j = 1, 2, . . . , w, we set Pj = s, [qj + 1], [qj + 2], . . . , [qj], t. So P1, P2, . . . , Pw are

mutually independent paths between s and t.

Case 2: (s, t) ∈ E(G). Assume that |S ∩ T | ≤ degG(s) + degG(t) − n + 2 − 1. We obtain

degG(s) + degG(t) = |S| + |T | = |S ∪ T | + |S ∩ T | ≤ degG(s) + degG(t) − n + 2 − 1 + n − 2. This is a contradiction. Thus, there are at leastw = degG(s)+degG(t)−n+2 elements in S∩T . Let q1, q2, . . . , qw

be the elements in S∩ T . For j = 1, 2, . . . , w, we set Pj = s, [qj + 1], [qj + 2], . . . , [qj], t, and P1, P2, . . . , Pw are mutually independent paths between s and t.

Example. Let G be the graph (K1∪ Kn−d−1) ∨ Kd where d is an integer with 4 ≤ d < n − 1. So ¯e = n − 1 − d ≤ n − 4. Let x be the vertex corresponding to K1, y be an arbitrary vertex in Kd, and z be a vertex in Kn−d−1. Then degG(x) = d, degG(y) = n − 1, degG(z) = n − 2, (x, y) ∈ E(G), (y, z) ∈ E(G), and (x, z) ∈ E(G). ByTheorem 6, there are n− 2 − ¯e = n − 2 − (n − 1 − d) = d − 1 mutually independent hamiltonian paths between any two distinct vertices of G. ByLemma 1, there are at most d− 1 mutually independent hamiltonian paths between x and y. Hence, the result inTheorem 6

is optimal.

Consider the same example as above; it is easy to check that any two vertices u and v in G, degG(u) + degG(v) ≥ n + 2. Let x and y be the same vertices as described above; by Theorem 7, there are deg_G(x) + deg_G(y) − n = d + (n − 1) − n = d − 1 mutually independent hamiltonian paths between x and y. ByLemma 1, there are at most d− 1 mutually independent hamiltonian paths between

x and y. Hence, the result inTheorem 7is also optimal.

4. Conjecture

Combining withTheorems 5and7, we have the following Corollary.

Corollary 1. Let r be a positive integer. Assume that G is a graph such that degG(x) +degG(y) ≥ n +r for any two distinct vertices x and y. Then there are at least r mutually independent hamiltonian paths between any two distinct vertices of G.

However, we would like to make the following conjecture. Suppose that r > 1 and G is a graph such that deg_G(u) + degG(v) ≥ n + r for any two distinct vertices u and v in G. Then there are at least r + 1 mutually independent hamiltonian paths between any two distinct vertices of G.

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