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Hamilton paths in generalized Petersen graphs
Discrete Mathematics 313 (2013) 1338–1341
Contents lists available at SciVerse ScienceDirect
Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
Hamilton paths in generalized Petersen graphs R. Bruce Richter Department of Combinatorics & Optimization, University of Waterloo, Canada
article
info
Article history: Received 3 June 2012 Received in revised form 25 February 2013 Accepted 28 February 2013 Available online 27 March 2013 Keywords: Hamilton paths Generalized Petersen graph
abstract Alspach and Qin proved that connected Cayley graphs of Hamiltonian groups (all subgroups are normal) are either Hamilton-connected (every pair of vertices is joined by a Hamilton path), or are bipartite and Hamilton-laceable (every pair on opposite sides of the bipartition are joined by a Hamilton path). Their proof made use of Hamilton-connectedness of certain Generalized Petersen graphs. In this work, we extend (and make a small correction to) the results of Alspach and Liu on Hamilton paths in generalized Petersen graphs. Alspach and Liu showed that, for k ∈ {1, 2, 3} and gcd(n, k) = 1, P (n, k) is either Hamilton-connected or bipartite and Hamilton-laceable, as long as (n, k) ̸= (6r + 5, 2) or (5, 3). For k = 2, we consider the remaining cases for n and completely determine which pairs of vertices in P (n, 2) are joined by Hamilton paths. However, the main point is to show that, for each k, it is a finite problem to determine, for all n, which pairs of vertices in P (n, k) are the ends of a Hamilton path. © 2013 Elsevier B.V. All rights reserved.
1. Introduction For positive integers n and k, with n ≥ 3, the generalized Petersen graph P (n, k) is the graph obtained from an n-cycle (u0 , u1 , . . . , un−1 , un ) by adding a pendant edge ui vi at each ui and then joining vi and vj if |j − i| ≡ k(mod n). The famous Petersen graph is P (5, 2). In general, P (n, k) = P (n, n − k), so the Petersen graph is also P (5, 3). These graphs have a long history; detailed information can be found in the book by Holton and Sheehan [4]. Motivation for us is found in the connection to the folklore conjecture that connected Cayley graphs are Hamiltonian (that is, have a cycle containing all the vertices). Alspach and Liu [1] give the history of showing that, except for the graphs P (6n + 5, 2), P (n, k) is Hamiltonian. Chen and Quimpo [3] proved that connected Cayley graphs of Hamilton groups (all subgroups are normal) have Hamilton cycles, while Cayley graphs of Abelian groups have the stronger property of being either Hamilton-connected (every pair of vertices is joined by a Hamilton path), or are bipartite and Hamilton-laceable (every pair on opposite sides of the bipartition are joined by a Hamilton path). Alspach and Qin [2] improved this by showing that Cayley graphs of Hamiltonian groups are either Hamilton-connected or bipartite and Hamilton-laceable. Alspach and Qin use the Hamilton-connection of certain generalized Petersen graphs in their proof. The general question we address here is: which P (n, k) are Hamilton-connected or -laceable? It is easy to see that P (n, k) is bipartite exactly when k is odd and n is even. Alspach and Liu [1] prove that, for k = 1, 2, 3, if gcd(n, k) = 1 and either n is odd or k = 2, then P (n, k) is Hamiltonconnected, except when k = 2 and n ≡ 5(mod 6) or k = 3 and n = 5, while if k = 1, 3, n is even, and gcd(n, k) = 1, then P (n, k) is Hamilton-laceable. They make the following remark: ‘‘The requirement that n and k are relatively prime . . . is not well understood. For example, there is no Hamilton path joining u1 and u3 in P (6, 2) but P (8, 2) is Hamilton-connected’’. (In fact P (8, 2) is not Hamilton-connected; this is an oversight that we discuss further in the next section.)
E-mail address: [email protected]. 0012-365X/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.disc.2013.02.021
R. Bruce Richter / Discrete Mathematics 313 (2013) 1338–1341
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Fig. 1. Extending the path from P (n, 2) to P (n + 6, 2).
In the first part of this work, we consider the cases of P (n, k) with k ≤ 3. For k ≤ 3 we fully determine which pairs of vertices in P (n, k) are joined by Hamilton paths. The techniques here are very similar to those of [1] and so will be only very lightly treated. We will emphasize the interesting point of proving that certain paths do not exist. Motivated by our considerations for k ≤ 3, in Section 3 we prove our main result. For any given k, there is an N so that it is enough to know which pairs of vertices are joined by Hamilton paths in all P (n, k), with n ≤ N. This knowledge then determines the existence of Hamilton paths in P (n, k) when n > N. I have been working with many students on this problem over many years. Vicki Mavraganis at Carleton University (1999), William Pensaert (2001), Yehua Wei (2006), and Loretta Vanderspek (2010), all from the University of Waterloo, have contributed directly to the results in this work. The last two in particular provided valuable insights for the main result mentioned in the preceding paragraph. 2. The cases k ≤ 3 If n is odd, then it is very easy (and known) to see that P (n, 1) is Hamilton-connected. If n is even, then P (n, 1) is easily seen to be Hamilton-laceable. It is worth noting that both these results are special cases of the theorem of Chen and Quimpo [3]: if G is a Cayley graph for an Abelian group, then either G is Hamilton-connected or G is bipartite and Hamilton-laceable. For k = 3, the following is proved in [1], completely solving the problem for k = 3. We will not provide another proof, except to say that the proof is the same kind of induction used to prove the existence of the Hamilton paths in all P (n, 2)’s. We also note that the graphs P (3r , 3), with r ≥ 3 and odd, have gcd(n, k) ̸= 1, and yet are Hamilton-connected. Theorem 2.1 ([1, Theorem 5.1]). If n is odd and at least 7, then P (n, 3) is Hamilton-connected. If n is even, at least 4 and not 6, then P (n, 3) is Hamilton-laceable. The remainder of this section is devoted to discussing the case k = 2. As in [1], the proof of the existence of Hamilton paths is inductive. This is accomplished by determining ways of extending a path in P (n, 2) to a path in P (n + 6, 2). We illustrate one example of this in Fig. 1. This method allows us to prove the following theorem. Theorem 2.2. Label the vertices of P (n, 2) so that (u0 , u1 , . . . , un−1 , u0 ) is an n-cycle, for i = 0, 1, . . . , n − 1, ui is also adjacent to vi , and vi and vi+2 are adjacent (all indices being read modulo n). Then: 1. P (n, 2) is Hamilton-connected if and only if n ≡ 1 or 3(mod 6). 2. If n ≡ 0(mod 6) and x and y are distinct vertices of P (n, 2) so that, for any i and t , {x, y} is neither of the pairs {ui , ui+2 } and {ui , ui+6t }, then there is a Hamilton path joining x and y in P (n, 2). 3. If n ≡ 2(mod 6) and {x, y} ̸= {vi , vi+4+6t }, for any i and t, then there is a Hamilton path joining x and y in P (n, 2). 4. If n ≡ 4(mod 6) and {x, y} is none of the pairs {ui , ui+2 }, {ui , vi±1 }, {ui , vi+2+6t }, {vi , vi+4+6t }, for any i and t, then there is a Hamilton path joining x and y in P (n, 2). 5. If n ≡ 5(mod 6), and x and y are not adjacent and {x, y} ̸= {vi , vi+3+6t } for any i and t, then there is a Hamilton path joining x and y in P (n, 2). We make two remarks. Firstly, Theorem 2.2 provides a complete description of which pairs of vertices in P (n, 2) are joined by Hamilton paths. This is different from [1, Theorem 4.4], that determines only whether or not P (n, 2) is Hamiltonconnected. Secondly, there is an error in [1, Theorem 4.4]. We gave earlier the quote that asserted P (8, 2) is Hamiltonconnected. This is incorrect. There is no Hamilton path from v0 to v4 ; up to rotational shifts, this is the one path missing in Theorem 2.2 (3). For the sake of completeness, here is an outline of a proof that P (8, 2) does not have a Hamilton path joining v0 and v4 . The 4-cycle (v0 , v2 , v4 , v6 , v0 ) contains both v0 and v4 . Each of v2 and v6 is incident with two edges of a Hamilton path H from v0 to v4 , while each of v0 and v4 is incident with only one. Thus, we may assume v0 v2 and v4 v6 are precisely the edges of the 4-cycle in H. This implies u7 u0 , u0 u1 , u3 u4 , and u4 u5 are also in H. The other 4-cycle with the odd-indexed vi ’s must also have at least two edges in H; it cannot have all four edges in H. If it has only two edges, then the four ui -edges we know are in H combine with these two edges and the four u2i+1 v2i+1 edges to make one or two cycles in H, obviously impossible. If three edges, then, so as not to make a cycle in H, up to symmetry it is v7 v1 , v1 v3 , and v3 v5 . But now u2 must be adjacent to all of u1 , u3 , and v2 in H, the final contradiction.
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Fig. 2. One of the even cases.
The proof of non-existence of a Hamilton path joining particular vertices is accomplished much as is done in the proof of [1, Theorem 4.4] to show that there is no Hamilton path from u0 to u2 in P (n, 2), when n ≡ 2 or 4(mod 6). We pick as an illustration the less specific example, when n ≡ 0(mod 6), that there is no Hamilton path joining v0 and v6t , for any t. We leave it to the reader to check the result is true in the base case n = 12: there is no Hamilton path joining v0 to v6 in P (12, 3). For the inductive step, suppose n ≥ 12 and that there is, for some t, a Hamilton path in P (n + 6, 2) joining v0 and v6t . We may suppose 6t ≤ n/2, in which case n + 6 − 6t ≥ 12. We set up a little notation to help describe the reduction process. For i, j ∈ {0, 1, . . . , n − 1}, we let Vi,j denote the set of vertices {uk , vk | k ∈ {i, i + 1, i + 2, . . . , i + j}} (all additions are in Zn ). The (i, j)-reduction of P (n, 2) is the graph G≻i,j obtained from P (n, 2) by deleting all the edges uk vk , k = i, i + 1, . . . , i + j − 1 and contracting all the edges uk−1 uk and vk−1 vk+1 , for the same range of k (that is, i to i + j − 1). This removes the vertices ui , . . . , ui+j−1 and vi , . . . , vi+j−1 from P (n, 2). We note that, when j is even, G≻i,j is isomorphic to P (n − j, 2). (When j is odd, then we end up with – in the new labelling – vi joined to vi+1 and vi−1 joined to vi+2 .) The distinction between G≻i,j and P (n − j, 2) is only in the names of the vertices; in G≻i,j they retain their labels from P (n, 2), making it easier to state our claims below. The other notion that we need is that of the cut Ci : for i ∈ {0, 1, . . . , n − 1}, this consists of the edges ui−1 ui , vi−2 vi , and vi−1 vi+1 . Claim 1. Let x, y ∈ {u, v} and suppose P is a Hamilton path joining x0 and yj in P (n, 2), with j ≤ n/2. If |Cj+1 ∩ P | is even and n − j ≥ 10, then there is a Hamilton path joining x0 and yj in G≻j+1,6 . Proof. We observe that, for k = j + 2, j + 3, . . . , n, Cj+1 ∪ Ck is an edge cut in P (n, 2) having both ends of P on the same side. This implies that |Cj+1 ∩ E (P )| and |Ck ∩ E (P )| have the same parity. Since |Cj+1 ∩ E (P )| is even, |Ck ∩ E (P )| is also even. There is at most one k for which Ck ∩ E (P ) can be empty. If there is such a k, then we know exactly how the path traverses the vertices incident with edges in Ck . By a simple adjustment to P, we can move the empty cut from Ck to Ck+3 ; thus, we can assume k ∈ {n − 2, n − 1, n}. Whether there is a k so that Ck ∩ E (P ) is empty or not, for k = j + 1, j + 2, . . . , j + 7, |Ck ∩ E (P )| = 2. There are three cases to check (which two edges are in Cj+1 ∩ E (P )), but each is easy. One is illustrated in Fig. 2. The conclusion is that the two edges in Cj+7 ∩ E (P ) are the +6 translates of those in Cj+1 ∩ E (P ). An important point in the argument is to realize that Cj+3 ∩ E (P ) also has two edges; the illustration shows the only way this can happen, given that Cj+1 ∩ E (P ) is as shown. Moreover, P has two subpaths among the vertices in Vj,j+7 . The edge of Cj+1 in one of those subpaths has its +6 translate in the same subpath. This property shows that G≻j+1,6 has a Hamilton path joining x0 and yj , as required. A more tedious version of the same style of argument produces the following. We point out that these arguments work because, once we know Ck ∩ E (P ), the nearby portion of P is determined. This allows us to make a move to a different Hamilton path P ′ having the same ends. If the intersection of a nearby cut with P is not determined, then we can consider the different possibilities (which are highly restricted) and either obtain a repetition or obtain what we want. Claim 2. Let x, y ∈ {u, v} and let P be a Hamilton path joining x0 and yj in P (n, 2), with j ≤ n/2. If |Cj+1 ∩ E (P )| is odd and n − j ≥ 12, then there is a Hamilton path joining x0 and yj in some G≻k,6 with k either j + 1 or j + 4. In the proof of this claim, one first aims to get a Hamilton path joining x0 and yj in P (n, 2) so that all the cuts Ck , either for k = j + 4, j + 5, . . . , n, or for k = j + 1, j + 2, . . . , n − 5 have all their edges in P. In these cases, we proceed as in the proof of Claim 1. The above works unless both Cj+1 and Cn have only one edge in P. In this remaining case, either we find that Cj+7 ∩ E (P ) has just one edge, and this is the +6 translate of the edge in Cj+1 ∩ E (P ) or we can arrange seven consecutive cuts in a row to have all edges in P. In the former case, we can easily reduce to G≻j+1,6 , while in the latter case we can reduce by 6 either at j + 1 or j + 4, depending on the details of P. 3. General k The arguments for non-existence in the case k = 2 are suggestive that there is a more general result that can be proved. It is not very surprising that a theorem of the following type holds.
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Theorem 3.1. Let k > 0 be an integer. Then there exist positive integers Nk and rk so that, if 1. n ≥ Nk and, 2. for x, y ∈ {u, v} and j ≤ n/2, there is a Hamilton path in P (n, k) joining x0 and yj , then there is a Hamilton path in P (n − rk k, k) joining x0 and yj and a Hamilton path in P (n + rk k, k) joining x0 and yj . The rest of this article is devoted to the proof of this theorem. Here is the basic strategy. For general k, the cut Ci now consists of k + 1 edges: the edge ui ui+1 and the k edges vi−t vi+k−t , for t = 0, 1, . . . , k − 1. For i = j + 1, j + 2, . . . , n, the cut Ci meets P in any number of edges from 0 to k + 1. The intersection of P with the subgraph of P (n, k) induced by Vj+1,i consists of paths and each path has two end vertices each incident with one edge of P that is also in Cj+1 ∪ Ci . These paths are the (j, i)-strands of P and, for each (j, i)-strand, the two edges in E (P ) ∩ (Cj+1 ∪ Ci ) are the links of the strand. For i, i′ ∈ {j + 1, j + 2, . . . , n}, the cuts Ci and Ci′ are P-congruent if there is a bijection between the (j, i)-strands of P and the (j, i′ )-strands of P, so that each link of a (j, i)-strand is either an edge of Cj+1 and that edge is a link of the corresponding (j, i′ )-strand or an edge of Ci and its (i′ − i)-translate in Ci′ is a link of the corresponding (j, i′ )-strand. For example, if a (j, i)-strand has a link e ∈ Cj+1 and a link f in Ci , then the corresponding (j, i′ )-strand is the unique (j, i′ )-strand that has e as a link, and the other link for this strand is the (i′ − i)-translate of f . The crucial observation is the following. Proposition 3.2. Let x, y ∈ {u, v} and j ≤ n/2 be such that there is a Hamilton path in P (n, k) joining x0 and yj . Suppose Ci and Ci′ are P-congruent cuts of P (n, k), with i, i′ ∈ {j + 2, j + 3, . . . , n}, i′ > i, and i′ − i a multiple of k. Then there is a Hamilton path in P (n − (i′ − i), k) joining x0 and yj and a Hamilton path in P (n + (i′ − i), k) joining x0 and yj . Proof. For the first, we simply apply the analogue of reduction to get G≻i,i′ −i ; this is isomorphic to P (n − (i′ − i), k) and has a Hamilton path joining x0 and yj . Because we are removing vertices with index larger than j, the labels x0 and yj are the same vertices in G≻i,i′ −i and P (n − (i′ − i), k). For the second, we subdivide ui ui+1 i′ − i times, inserting the vertices (u′1 , u′2 , . . . , u′i′ −i ) between ui and ui+1 . Each u′r is adjacent to a new vertex vr′ . We cut the edges vs vs+k , for s = j − k + 1, j − k + 2, . . . , j and then add edges back in to make an isomorph of P (n + (i′ − i), k). (So, for example, we have vj − k + r adjacent to vr′ and vj+1+r is adjacent to vi′′ −i−k+r .) We now copy the (i, i′ )-strands of P into the new vertices u′r and vr′ , to obtain the desired Hamilton path in P (n + (i′ − i), k). This proposition is not quite enough to prove Theorem 3.1, but the rest is just arithmetic and the Pigeonhole Principle. For each r > 0, there are only finitely many possibilities for how the (j + 1, j + 1 + rk)-strands and links interact with the cuts Cj and Cj+1+rk ; let ak denote the number of these different possibilities. Thus, if n − j − 1 > ak k, then there must be distinct r1 , r2 ∈ {1, 2, . . . , ak + 1} so that the two cuts Cj+1+r1 k and Cj+1+r2 k are P-congruent. For any positive integer ℓ, if n − j − 1 ≥ ℓ(ak + 1)k, there are at least ℓ pairs (r1 , r1′ ), (r2 , r2′ ), . . . , (rℓ , rℓ′ ) so that 1 ≤ r1 < r1′ ≤ ak + 1 < r2 < r2′ ≤ 2(ak + 1) < · · · < rℓ , rℓ′ ≤ ℓ(ak + 1) and, for each i = 1, 2, . . . , ℓ, Cj+1+ri k and Cj+1+r ′ k are P-congruent. It is evident that, for each k, rk′ − rk ≤ ak , so there are ak possibilities for the difference rk′ − rk . i
Let Mk = lcm{1, 2, . . . , ak }. Finally, choose ℓ = Mk ak . This gives Mk ak pairs (ri , ri′ ) as in the preceding paragraph. Thus, among the Mk ak pairs (ri , ri′ ), the same difference ri′ − ri occurs at least Mk times. Let d be such a difference that occurs at least Mk times and let q = (Mk /d)ak . Clearly q ≤ Mk . Let i1 < i2 < · · · < iq be indices so that, for each s = 1, 2, . . . , q, ri′s − ris = d. Apply Proposition 3.2 on each of these q pairs (ris , ri′s ) in each direction to see that there is a Hamilton path joining x0 and yj in each of P (n + Mk ak (ak + 1)k, k) and P (n − Mk ak (ak + 1)k, k), as required to prove Theorem 3.1. References [1] B. Alspach, J. Liu, On the Hamilton connectivity of generalized Petersen graphs, Discrete Math. 309 (17) (2009) 5461–5473. [2] B. Alspach, Y. Qin, Hamilton-connected Cayley graphs on Hamiltonian groups, European J. Combin. 22 (22) (2001) 777–787. [3] C.C. Chen, N. Quimpo, On strongly hamiltonian abelian group graphs, in: Combinatorial Mathematics VIII, in: Lecture Notes in Mathematics, vol. 884, Springer-Verlag, 1981, pp. 23–34. [4] The Petersen Graph, in: Australian Mathematical Society Lecture Series, vol. 7, Cambridge University Press, Cambridge, 1993.
Contents lists available at SciVerse ScienceDirect
Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
Hamilton paths in generalized Petersen graphs R. Bruce Richter Department of Combinatorics & Optimization, University of Waterloo, Canada
article
info
Article history: Received 3 June 2012 Received in revised form 25 February 2013 Accepted 28 February 2013 Available online 27 March 2013 Keywords: Hamilton paths Generalized Petersen graph
abstract Alspach and Qin proved that connected Cayley graphs of Hamiltonian groups (all subgroups are normal) are either Hamilton-connected (every pair of vertices is joined by a Hamilton path), or are bipartite and Hamilton-laceable (every pair on opposite sides of the bipartition are joined by a Hamilton path). Their proof made use of Hamilton-connectedness of certain Generalized Petersen graphs. In this work, we extend (and make a small correction to) the results of Alspach and Liu on Hamilton paths in generalized Petersen graphs. Alspach and Liu showed that, for k ∈ {1, 2, 3} and gcd(n, k) = 1, P (n, k) is either Hamilton-connected or bipartite and Hamilton-laceable, as long as (n, k) ̸= (6r + 5, 2) or (5, 3). For k = 2, we consider the remaining cases for n and completely determine which pairs of vertices in P (n, 2) are joined by Hamilton paths. However, the main point is to show that, for each k, it is a finite problem to determine, for all n, which pairs of vertices in P (n, k) are the ends of a Hamilton path. © 2013 Elsevier B.V. All rights reserved.
1. Introduction For positive integers n and k, with n ≥ 3, the generalized Petersen graph P (n, k) is the graph obtained from an n-cycle (u0 , u1 , . . . , un−1 , un ) by adding a pendant edge ui vi at each ui and then joining vi and vj if |j − i| ≡ k(mod n). The famous Petersen graph is P (5, 2). In general, P (n, k) = P (n, n − k), so the Petersen graph is also P (5, 3). These graphs have a long history; detailed information can be found in the book by Holton and Sheehan [4]. Motivation for us is found in the connection to the folklore conjecture that connected Cayley graphs are Hamiltonian (that is, have a cycle containing all the vertices). Alspach and Liu [1] give the history of showing that, except for the graphs P (6n + 5, 2), P (n, k) is Hamiltonian. Chen and Quimpo [3] proved that connected Cayley graphs of Hamilton groups (all subgroups are normal) have Hamilton cycles, while Cayley graphs of Abelian groups have the stronger property of being either Hamilton-connected (every pair of vertices is joined by a Hamilton path), or are bipartite and Hamilton-laceable (every pair on opposite sides of the bipartition are joined by a Hamilton path). Alspach and Qin [2] improved this by showing that Cayley graphs of Hamiltonian groups are either Hamilton-connected or bipartite and Hamilton-laceable. Alspach and Qin use the Hamilton-connection of certain generalized Petersen graphs in their proof. The general question we address here is: which P (n, k) are Hamilton-connected or -laceable? It is easy to see that P (n, k) is bipartite exactly when k is odd and n is even. Alspach and Liu [1] prove that, for k = 1, 2, 3, if gcd(n, k) = 1 and either n is odd or k = 2, then P (n, k) is Hamiltonconnected, except when k = 2 and n ≡ 5(mod 6) or k = 3 and n = 5, while if k = 1, 3, n is even, and gcd(n, k) = 1, then P (n, k) is Hamilton-laceable. They make the following remark: ‘‘The requirement that n and k are relatively prime . . . is not well understood. For example, there is no Hamilton path joining u1 and u3 in P (6, 2) but P (8, 2) is Hamilton-connected’’. (In fact P (8, 2) is not Hamilton-connected; this is an oversight that we discuss further in the next section.)
E-mail address: [email protected]. 0012-365X/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.disc.2013.02.021
R. Bruce Richter / Discrete Mathematics 313 (2013) 1338–1341
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Fig. 1. Extending the path from P (n, 2) to P (n + 6, 2).
In the first part of this work, we consider the cases of P (n, k) with k ≤ 3. For k ≤ 3 we fully determine which pairs of vertices in P (n, k) are joined by Hamilton paths. The techniques here are very similar to those of [1] and so will be only very lightly treated. We will emphasize the interesting point of proving that certain paths do not exist. Motivated by our considerations for k ≤ 3, in Section 3 we prove our main result. For any given k, there is an N so that it is enough to know which pairs of vertices are joined by Hamilton paths in all P (n, k), with n ≤ N. This knowledge then determines the existence of Hamilton paths in P (n, k) when n > N. I have been working with many students on this problem over many years. Vicki Mavraganis at Carleton University (1999), William Pensaert (2001), Yehua Wei (2006), and Loretta Vanderspek (2010), all from the University of Waterloo, have contributed directly to the results in this work. The last two in particular provided valuable insights for the main result mentioned in the preceding paragraph. 2. The cases k ≤ 3 If n is odd, then it is very easy (and known) to see that P (n, 1) is Hamilton-connected. If n is even, then P (n, 1) is easily seen to be Hamilton-laceable. It is worth noting that both these results are special cases of the theorem of Chen and Quimpo [3]: if G is a Cayley graph for an Abelian group, then either G is Hamilton-connected or G is bipartite and Hamilton-laceable. For k = 3, the following is proved in [1], completely solving the problem for k = 3. We will not provide another proof, except to say that the proof is the same kind of induction used to prove the existence of the Hamilton paths in all P (n, 2)’s. We also note that the graphs P (3r , 3), with r ≥ 3 and odd, have gcd(n, k) ̸= 1, and yet are Hamilton-connected. Theorem 2.1 ([1, Theorem 5.1]). If n is odd and at least 7, then P (n, 3) is Hamilton-connected. If n is even, at least 4 and not 6, then P (n, 3) is Hamilton-laceable. The remainder of this section is devoted to discussing the case k = 2. As in [1], the proof of the existence of Hamilton paths is inductive. This is accomplished by determining ways of extending a path in P (n, 2) to a path in P (n + 6, 2). We illustrate one example of this in Fig. 1. This method allows us to prove the following theorem. Theorem 2.2. Label the vertices of P (n, 2) so that (u0 , u1 , . . . , un−1 , u0 ) is an n-cycle, for i = 0, 1, . . . , n − 1, ui is also adjacent to vi , and vi and vi+2 are adjacent (all indices being read modulo n). Then: 1. P (n, 2) is Hamilton-connected if and only if n ≡ 1 or 3(mod 6). 2. If n ≡ 0(mod 6) and x and y are distinct vertices of P (n, 2) so that, for any i and t , {x, y} is neither of the pairs {ui , ui+2 } and {ui , ui+6t }, then there is a Hamilton path joining x and y in P (n, 2). 3. If n ≡ 2(mod 6) and {x, y} ̸= {vi , vi+4+6t }, for any i and t, then there is a Hamilton path joining x and y in P (n, 2). 4. If n ≡ 4(mod 6) and {x, y} is none of the pairs {ui , ui+2 }, {ui , vi±1 }, {ui , vi+2+6t }, {vi , vi+4+6t }, for any i and t, then there is a Hamilton path joining x and y in P (n, 2). 5. If n ≡ 5(mod 6), and x and y are not adjacent and {x, y} ̸= {vi , vi+3+6t } for any i and t, then there is a Hamilton path joining x and y in P (n, 2). We make two remarks. Firstly, Theorem 2.2 provides a complete description of which pairs of vertices in P (n, 2) are joined by Hamilton paths. This is different from [1, Theorem 4.4], that determines only whether or not P (n, 2) is Hamiltonconnected. Secondly, there is an error in [1, Theorem 4.4]. We gave earlier the quote that asserted P (8, 2) is Hamiltonconnected. This is incorrect. There is no Hamilton path from v0 to v4 ; up to rotational shifts, this is the one path missing in Theorem 2.2 (3). For the sake of completeness, here is an outline of a proof that P (8, 2) does not have a Hamilton path joining v0 and v4 . The 4-cycle (v0 , v2 , v4 , v6 , v0 ) contains both v0 and v4 . Each of v2 and v6 is incident with two edges of a Hamilton path H from v0 to v4 , while each of v0 and v4 is incident with only one. Thus, we may assume v0 v2 and v4 v6 are precisely the edges of the 4-cycle in H. This implies u7 u0 , u0 u1 , u3 u4 , and u4 u5 are also in H. The other 4-cycle with the odd-indexed vi ’s must also have at least two edges in H; it cannot have all four edges in H. If it has only two edges, then the four ui -edges we know are in H combine with these two edges and the four u2i+1 v2i+1 edges to make one or two cycles in H, obviously impossible. If three edges, then, so as not to make a cycle in H, up to symmetry it is v7 v1 , v1 v3 , and v3 v5 . But now u2 must be adjacent to all of u1 , u3 , and v2 in H, the final contradiction.
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Fig. 2. One of the even cases.
The proof of non-existence of a Hamilton path joining particular vertices is accomplished much as is done in the proof of [1, Theorem 4.4] to show that there is no Hamilton path from u0 to u2 in P (n, 2), when n ≡ 2 or 4(mod 6). We pick as an illustration the less specific example, when n ≡ 0(mod 6), that there is no Hamilton path joining v0 and v6t , for any t. We leave it to the reader to check the result is true in the base case n = 12: there is no Hamilton path joining v0 to v6 in P (12, 3). For the inductive step, suppose n ≥ 12 and that there is, for some t, a Hamilton path in P (n + 6, 2) joining v0 and v6t . We may suppose 6t ≤ n/2, in which case n + 6 − 6t ≥ 12. We set up a little notation to help describe the reduction process. For i, j ∈ {0, 1, . . . , n − 1}, we let Vi,j denote the set of vertices {uk , vk | k ∈ {i, i + 1, i + 2, . . . , i + j}} (all additions are in Zn ). The (i, j)-reduction of P (n, 2) is the graph G≻i,j obtained from P (n, 2) by deleting all the edges uk vk , k = i, i + 1, . . . , i + j − 1 and contracting all the edges uk−1 uk and vk−1 vk+1 , for the same range of k (that is, i to i + j − 1). This removes the vertices ui , . . . , ui+j−1 and vi , . . . , vi+j−1 from P (n, 2). We note that, when j is even, G≻i,j is isomorphic to P (n − j, 2). (When j is odd, then we end up with – in the new labelling – vi joined to vi+1 and vi−1 joined to vi+2 .) The distinction between G≻i,j and P (n − j, 2) is only in the names of the vertices; in G≻i,j they retain their labels from P (n, 2), making it easier to state our claims below. The other notion that we need is that of the cut Ci : for i ∈ {0, 1, . . . , n − 1}, this consists of the edges ui−1 ui , vi−2 vi , and vi−1 vi+1 . Claim 1. Let x, y ∈ {u, v} and suppose P is a Hamilton path joining x0 and yj in P (n, 2), with j ≤ n/2. If |Cj+1 ∩ P | is even and n − j ≥ 10, then there is a Hamilton path joining x0 and yj in G≻j+1,6 . Proof. We observe that, for k = j + 2, j + 3, . . . , n, Cj+1 ∪ Ck is an edge cut in P (n, 2) having both ends of P on the same side. This implies that |Cj+1 ∩ E (P )| and |Ck ∩ E (P )| have the same parity. Since |Cj+1 ∩ E (P )| is even, |Ck ∩ E (P )| is also even. There is at most one k for which Ck ∩ E (P ) can be empty. If there is such a k, then we know exactly how the path traverses the vertices incident with edges in Ck . By a simple adjustment to P, we can move the empty cut from Ck to Ck+3 ; thus, we can assume k ∈ {n − 2, n − 1, n}. Whether there is a k so that Ck ∩ E (P ) is empty or not, for k = j + 1, j + 2, . . . , j + 7, |Ck ∩ E (P )| = 2. There are three cases to check (which two edges are in Cj+1 ∩ E (P )), but each is easy. One is illustrated in Fig. 2. The conclusion is that the two edges in Cj+7 ∩ E (P ) are the +6 translates of those in Cj+1 ∩ E (P ). An important point in the argument is to realize that Cj+3 ∩ E (P ) also has two edges; the illustration shows the only way this can happen, given that Cj+1 ∩ E (P ) is as shown. Moreover, P has two subpaths among the vertices in Vj,j+7 . The edge of Cj+1 in one of those subpaths has its +6 translate in the same subpath. This property shows that G≻j+1,6 has a Hamilton path joining x0 and yj , as required. A more tedious version of the same style of argument produces the following. We point out that these arguments work because, once we know Ck ∩ E (P ), the nearby portion of P is determined. This allows us to make a move to a different Hamilton path P ′ having the same ends. If the intersection of a nearby cut with P is not determined, then we can consider the different possibilities (which are highly restricted) and either obtain a repetition or obtain what we want. Claim 2. Let x, y ∈ {u, v} and let P be a Hamilton path joining x0 and yj in P (n, 2), with j ≤ n/2. If |Cj+1 ∩ E (P )| is odd and n − j ≥ 12, then there is a Hamilton path joining x0 and yj in some G≻k,6 with k either j + 1 or j + 4. In the proof of this claim, one first aims to get a Hamilton path joining x0 and yj in P (n, 2) so that all the cuts Ck , either for k = j + 4, j + 5, . . . , n, or for k = j + 1, j + 2, . . . , n − 5 have all their edges in P. In these cases, we proceed as in the proof of Claim 1. The above works unless both Cj+1 and Cn have only one edge in P. In this remaining case, either we find that Cj+7 ∩ E (P ) has just one edge, and this is the +6 translate of the edge in Cj+1 ∩ E (P ) or we can arrange seven consecutive cuts in a row to have all edges in P. In the former case, we can easily reduce to G≻j+1,6 , while in the latter case we can reduce by 6 either at j + 1 or j + 4, depending on the details of P. 3. General k The arguments for non-existence in the case k = 2 are suggestive that there is a more general result that can be proved. It is not very surprising that a theorem of the following type holds.
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Theorem 3.1. Let k > 0 be an integer. Then there exist positive integers Nk and rk so that, if 1. n ≥ Nk and, 2. for x, y ∈ {u, v} and j ≤ n/2, there is a Hamilton path in P (n, k) joining x0 and yj , then there is a Hamilton path in P (n − rk k, k) joining x0 and yj and a Hamilton path in P (n + rk k, k) joining x0 and yj . The rest of this article is devoted to the proof of this theorem. Here is the basic strategy. For general k, the cut Ci now consists of k + 1 edges: the edge ui ui+1 and the k edges vi−t vi+k−t , for t = 0, 1, . . . , k − 1. For i = j + 1, j + 2, . . . , n, the cut Ci meets P in any number of edges from 0 to k + 1. The intersection of P with the subgraph of P (n, k) induced by Vj+1,i consists of paths and each path has two end vertices each incident with one edge of P that is also in Cj+1 ∪ Ci . These paths are the (j, i)-strands of P and, for each (j, i)-strand, the two edges in E (P ) ∩ (Cj+1 ∪ Ci ) are the links of the strand. For i, i′ ∈ {j + 1, j + 2, . . . , n}, the cuts Ci and Ci′ are P-congruent if there is a bijection between the (j, i)-strands of P and the (j, i′ )-strands of P, so that each link of a (j, i)-strand is either an edge of Cj+1 and that edge is a link of the corresponding (j, i′ )-strand or an edge of Ci and its (i′ − i)-translate in Ci′ is a link of the corresponding (j, i′ )-strand. For example, if a (j, i)-strand has a link e ∈ Cj+1 and a link f in Ci , then the corresponding (j, i′ )-strand is the unique (j, i′ )-strand that has e as a link, and the other link for this strand is the (i′ − i)-translate of f . The crucial observation is the following. Proposition 3.2. Let x, y ∈ {u, v} and j ≤ n/2 be such that there is a Hamilton path in P (n, k) joining x0 and yj . Suppose Ci and Ci′ are P-congruent cuts of P (n, k), with i, i′ ∈ {j + 2, j + 3, . . . , n}, i′ > i, and i′ − i a multiple of k. Then there is a Hamilton path in P (n − (i′ − i), k) joining x0 and yj and a Hamilton path in P (n + (i′ − i), k) joining x0 and yj . Proof. For the first, we simply apply the analogue of reduction to get G≻i,i′ −i ; this is isomorphic to P (n − (i′ − i), k) and has a Hamilton path joining x0 and yj . Because we are removing vertices with index larger than j, the labels x0 and yj are the same vertices in G≻i,i′ −i and P (n − (i′ − i), k). For the second, we subdivide ui ui+1 i′ − i times, inserting the vertices (u′1 , u′2 , . . . , u′i′ −i ) between ui and ui+1 . Each u′r is adjacent to a new vertex vr′ . We cut the edges vs vs+k , for s = j − k + 1, j − k + 2, . . . , j and then add edges back in to make an isomorph of P (n + (i′ − i), k). (So, for example, we have vj − k + r adjacent to vr′ and vj+1+r is adjacent to vi′′ −i−k+r .) We now copy the (i, i′ )-strands of P into the new vertices u′r and vr′ , to obtain the desired Hamilton path in P (n + (i′ − i), k). This proposition is not quite enough to prove Theorem 3.1, but the rest is just arithmetic and the Pigeonhole Principle. For each r > 0, there are only finitely many possibilities for how the (j + 1, j + 1 + rk)-strands and links interact with the cuts Cj and Cj+1+rk ; let ak denote the number of these different possibilities. Thus, if n − j − 1 > ak k, then there must be distinct r1 , r2 ∈ {1, 2, . . . , ak + 1} so that the two cuts Cj+1+r1 k and Cj+1+r2 k are P-congruent. For any positive integer ℓ, if n − j − 1 ≥ ℓ(ak + 1)k, there are at least ℓ pairs (r1 , r1′ ), (r2 , r2′ ), . . . , (rℓ , rℓ′ ) so that 1 ≤ r1 < r1′ ≤ ak + 1 < r2 < r2′ ≤ 2(ak + 1) < · · · < rℓ , rℓ′ ≤ ℓ(ak + 1) and, for each i = 1, 2, . . . , ℓ, Cj+1+ri k and Cj+1+r ′ k are P-congruent. It is evident that, for each k, rk′ − rk ≤ ak , so there are ak possibilities for the difference rk′ − rk . i
Let Mk = lcm{1, 2, . . . , ak }. Finally, choose ℓ = Mk ak . This gives Mk ak pairs (ri , ri′ ) as in the preceding paragraph. Thus, among the Mk ak pairs (ri , ri′ ), the same difference ri′ − ri occurs at least Mk times. Let d be such a difference that occurs at least Mk times and let q = (Mk /d)ak . Clearly q ≤ Mk . Let i1 < i2 < · · · < iq be indices so that, for each s = 1, 2, . . . , q, ri′s − ris = d. Apply Proposition 3.2 on each of these q pairs (ris , ri′s ) in each direction to see that there is a Hamilton path joining x0 and yj in each of P (n + Mk ak (ak + 1)k, k) and P (n − Mk ak (ak + 1)k, k), as required to prove Theorem 3.1. References [1] B. Alspach, J. Liu, On the Hamilton connectivity of generalized Petersen graphs, Discrete Math. 309 (17) (2009) 5461–5473. [2] B. Alspach, Y. Qin, Hamilton-connected Cayley graphs on Hamiltonian groups, European J. Combin. 22 (22) (2001) 777–787. [3] C.C. Chen, N. Quimpo, On strongly hamiltonian abelian group graphs, in: Combinatorial Mathematics VIII, in: Lecture Notes in Mathematics, vol. 884, Springer-Verlag, 1981, pp. 23–34. [4] The Petersen Graph, in: Australian Mathematical Society Lecture Series, vol. 7, Cambridge University Press, Cambridge, 1993.