Abstract
A spanning path in a graph G is called a Hamiltonian path. To determine which graphs possess such paths is an NP-complete problem. A graph G is called Hamiltonian-connected if any two vertices of G are connected by a Hamiltonian path. We consider here the family of Toeplitz graphs. About them, it is known only for that is Hamiltonian-connected, while some particular cases of for and have also been investigated regarding Hamiltonian connectedness. Here, we prove that the nonbipartite Toeplitz graph is Hamiltonian-connected for all and .
1. Introduction
A path in a finite undirected graph G is called a Hamiltonian path if it visits each vertex of G exactly once. We call the graph G Hamiltonian-connected if for any pair of distinct vertices x and y of G, there exists a Hamiltonian path from x to y. In 1963, Ore introduced the family of Hamiltonian-connected graphs [13]. The Hamiltonian path problem, is the computational complexity problem of finding Hamiltonian paths in graphs, and related graphs are among the most famous NP-complete problems, see [14]. In this paper, we are investigating this property of Hamiltonian connectedness for some classes of Toeplitz graphs.
Let such that . An undirected Toeplitz graph is a symmetric graph with the vertex set and with an edge between the vertices i and j if and only if for some . The integers are called entries or jumps. The adjacency matrix of any such graph is a symmetric Toeplitz matrix. Toeplitz graphs were introduced by G. Sierksma. The undirected Toeplitz graphs were first investigated by van Dal et al. [14] with respect to hamiltonicity. Heuberger [8] extended this study in 2002, while the directed case was studied in [9–11]. For results regarding different properties of Toeplitz graphs such as connectivity, bipartiteness, planarity, and colourability, see [3–8]. The well-known circulant graphs are particular cases of Toeplitz graphs. In fact, for each Toeplitz graph T, there exists a circulant graph C such that T is a spanning subgraph of C.
For , a circulant graph is a regular graph of degree or with the vertex set , in which two vertices i and j are adjacent if and only if , for some . Circulant graphs are Cayley graphs on the abelian group , i.e., the circulant graph is the Cayley graph . Furthermore, note that and (a cycle or cyclic graph on n vertices).
For results regarding connectivity of the Toeplitz graph, see [14], where it is shown that the graph has at least components. Therefore, for , the corresponding Toeplitz graph is disconnected. But, one may find a disconnected graph even for , e.g., .
In [11], it has been proven that is Hamiltonian-connected only for , while is Hamiltonian-connected for all values of n and s. The present paper is a sequel of [12], where it is proved that is Hamiltonian-connected only for . Thereafter, the case was considered and it was shown that the graphs and are Hamiltonian-connected. Here, we are presenting a more general result about with under the assumption that and are not both odd because otherwise the corresponding Toeplitz graph becomes bipartite, hence not Hamiltonian-connected [12]. A Toeplitz graph becomes circulant if for each entry , also occurs as an entry for all , see [1]. In this special class of Toeplitz graphs, we prove here the existence of Hamiltonian-connected graphs.
The following results are needed to prove our first result.
Theorem 1 (see [2]). A connected Cayley graph on an abelian group is Hamiltonian-connected if and only if it is neither cyclic nor bipartite.
Theorem 2 (see [15]). The circulant graph is connected if and only if .
Theorem 3 (see [7]). A connected circulant graph is bipartite if and only if are odd and n is even.
2. Main Results
We start with the following result which is a consequence of Theorem 1.
Theorem 4. If n is an odd integer and such that and , then is Hamiltonian-connected.
Proof. Since , under the given conditions and by Theorems 2 and 3, T is a connected noncyclic and nonbipartite Cayley graph Hence, it is Hamiltonian-connected by Theorem 1.
To prove our next main results, we need the following notation and lemmas.
Let T be a Toeplitz graph and be two vertices of T. The symbols and stand for the paths and respectively. By , we mean a path from p to with the set of vertices , and by , we mean a path from q to with the same vertices. Note that the existence of or is not guaranteed. Furthermore, it is easy to observe that if T is a Toeplitz graph of order n and there exists a path in T, then by the symmetry of Toeplitz graphs, there exists another path in T.
Lemma 1. If t is an even integer with or , then admits a Hamiltonian path from 1 to 2.
Proof. For the Toeplitz graph with t even admits a unique Hamiltonian path which is starting from 1, passing through the edge , and ending at 2. We use these paths as basic paths to construct our desired path in which are defined as follows:
When When for some ,When for some ,See also Figures 1(a)–1(c), respectively, for the illustration of , and
Now, by using , and we construct a Hamiltonian path from 1 to 2 in as follows.
When , a desired path obtained by using P is shown in Figure 2(a).
When for some , we use P and to get a suitable path shown in Figure 2(b).
Finally, when for some , we consider P and to obtain a path given in Figure 2(c) as desired.
(a)
(b)
(c)
Lemma 2. If t is an odd integer and n is an even integer, then admits a Hamiltonian path from 1 to 2.
Proof. Let be an even integer for some . For , consider the path shown in Figure 2(a), while for other values of i, follow the path shown in Figure 2(b).
Immediate consequences of Lemmas 1 and 2 are as follows.
Corollary 1. Let t be an even integer and be any two vertices of . Then, paths and exist in T if or .
Corollary 2. Let t be an odd integer and be any two vertices of . Then, paths and exist in T if or is odd.
Lemma 3. Let t be an even integer and x be any vertex of . Then, there exists a Hamiltonian path from x to in T.
Proof. For , the result is trivial. Paths for other values of x are listed in Table 1.
Now, by using Corollary 1 and Lemma 3, we prove our next lemma.
Lemma 4. Let be any two vertices of , where t is an even integer and . Then, admits a Hamiltonian path from x to y, except from 2 to (by symmetry, another one from to ).
Proof. Let be a Toeplitz graph with t even and . Because of symmetry of Toeplitz graphs, it suffices to show that T admits a Hamiltonian path from any vertex to each vertex of T. Take , any two vertices of T, other than the pair of vertices. We split our proof into two cases.
Case 1.
Let , then by Corollary 1, we have paths and in T. By joining and to the remaining subgraph of T, we obtain desired Hamiltonian paths and in T, respectively, for and . For illustration, see Figures 3(a) and 3(b), respectively.
Finally, to obtain Hamiltonian paths for remaining values of x, we assume , where and are integers for some such that . By applying Corollary 1 to T, we get and in and construct a desired path from x to in T. See also Figure 3(c).
(a)
(b)
(c)
Case 2. .
Here, we partition the vertex set of T into 5 subsets of vertices, according to Figure 4, and consider the following two subcases.
Case 3.
Corollary 1 guarantees the existence of for any vertex in T. Here, by using it, we are getting Hamiltonian paths from x to y in
When we consider the path from 1 to any vertex y in T.
If then possible Hamiltonian paths for different values of x and y are listed in Tables 2–4.
Case 4. and
The case when and is symmetric to the case when and . Hence, we remain only with the case when and . Here, by Lemma 3, we have a Hamiltonian path in from any to . By symmetry, another Hamiltonian path exists from to , of vertex set . By joining to by the paths we get a Hamiltonian path from x to y in T.
This completes the proof.
Now, by using the fact and Lemma 4, we prove our next main results.
Theorem 5. If t is an even integer, then the Toeplitz graph is hamiltonian-connected for all .
Proof. Let be a Toeplitz graph with t even and . Then, because of Lemma 4, we only have to establish the existence of a Hamiltonian path from 2 to . For this, we use the entry s along with other two entries 1 and t.
There are five cases to consider. In first four cases, we use to construct desired paths which exist for any vertex of T due to Corollary 1.(i)For , considered path is(ii)When a possible path is(iii); in this case, constructed path is(iv); a desired Hamiltonian path is(v): here, first, by using Lemma 3, we construct a path in from to , of vertex set . Then, by joining this path to the remaining subgraph of T, we get a Hamiltonian pathfrom 2 to . This concludes the proof.
Theorem 6. If t is odd and s is even, then the Toeplitz graph is Hamiltonian-connected for all .
Proof. Again by virtue of Lemma 4, for s even and we only need to prove the existence of a Hamiltonian path starting from 2 and ending at . Here, we consider the following four cases:(i); by Corollary 1, we have in , which helps us to get a desired path:(ii); again by applying Corollary 1 to , we get to construct a Hamiltonian path:(iii); here, is a desired path, which is constructed by using , obtained by applying Corollaries 1 and 2 to .(iv); in this case, we use Corollaries 1 and 2 to obtain in and in , respectively, which enables us to obtain a Hamiltonian pathfrom 2 to in T. This completes the proof.
3. Conclusion
We proved here the existence of a number N such that for , every nonbipartite Toeplitz graph is Hamiltonian-connected. Also, the family of Toeplitz graphs, which are also circulant, contains Hamiltonian-connected graphs.
Data Availability
Research data have been provided in the manuscript.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgements
The first author gratefully acknowledges financial support by the Deanship of Scientific Research, King Faisal University, through the Nasher track under the under grant no. 186236. The third author gratefully acknowledges financial support by NSF of China (no. 11871192) and the Program for Foreign experts of Hebei Province (no. 2019YX002A).