Hamiltonian cycles in circulant digraphs with jumps 2, 3, c [We need a real title???]

Transcription

1 Hamiltonian cycles in circulant digraphs with jumps,, c [We need a real title???] Abstract [We need an abstract???] 1 Introduction It is not known which circulant digraphs have hamiltonian cycles; this is a fundamental open question. However, the circulants of outdegree are the smallest ones that need to be considered, because a classic result of R. A. Rankin (see Theorem.7) provides a nice characterization for circulants of outdegree (and any strongly connected digraph of outdegree 1 is obviously hamiltonian). S. C. Locke and D. Witte [1] found two infinite families of non-hamiltonian circulant digraphs of outdegree. One of the families includes the following examples, which require introducing a piece of notation. Notation 1.1 For S Z, we use Circ(n; S) to denote the circulant digraph whose vertex set is Z n, and with an arc from v to v + s for each v Z n and s S. Theorem 1. (Locke-Witte, cf. [1, Thm. 1.4]) (1) Circ(6m;,, m + ) is not hamiltonian if and only if m is even. () Circ(6m;,, m + ) is not hamiltonian if and only if m is odd. In this paper, we completely characterize which loopless digraphs of the form Circ(n;,, c) that have outdegree are hamiltonian: Theorem 1. Assume c 0,, (mod n). The digraph Circ(n;,, c) is not hamiltonian iff (1) n is a multiple of 6, so we may write n = 6m, () either c = m + or c = m +, and () c is even. Preprint submitted to Elsevier Science 8 September 006

2 The direction ( ) of Theorem 1. is a restatement of part of the Locke-Witte Theorem (1.), so we need only prove the opposite direction. Acknowledgements The work of D.W.M. and J.M. was partially supported by research grants from Canada s National Science and Engineering Research Council. Preliminaries Our goal is to establish Theorem 1.( ). We will prove the contrapositive. Assumption.1 Throughout the paper: (1) We assume the situation of Theorem 1., so n, c Z +, and c 0,, (mod n). () We may assume c 1, 1 (mod n). (Otherwise, Circ(n;,, c) has a hamiltonian cycle consisting entirely of arcs of length c.) () Since the vertices of Circ(n;,, c) are elements of Z n, we may assume < c < n. (4) We assume n is divisible by 6. (Otherwise, Circ(n;,, c) has either a hamiltonian cycle consisting entirely of arcs of length or a hamiltonian cycle consisting entirely of arcs of length.) (5) We write n = 6m. Notation. Let H be a subdigraph of Circ(n;,, c), and let v be a vertex of H. (1) We let d + H(v) and d H(v) denote the number of arcs of H directed out of, and into, vertex v, respectively. () If d + H(v) = 1, and the arc from v to v + a is in H, then we say that v travels by a in H. Notation. Let u, w be integers representing vertices of Circ(n;,, c). If u 1 w < u + n, let I(u, w) = {u, u + 1,..., w} be the interval of vertices from u to w. (Note that I(u, u) = {u} and I(u, u 1) =.) Notation.4 For v 1, v Z n and s {,, c }, we use v s 1 v to denote the arc from v 1 to v 1 + s = v, and v s 1 v to denote a path of the form v 1, v 1 + s, v 1 + s,..., v 1 + ks = v.

3 We now treat two simple cases that do not conform to the more general structures that we deal with in later sections. Lemma.5 The digraph Circ(6m;,, 6m ) has a hamiltonian cycle. PROOF. For 0 i m 1, define the path Q 6i as follows: 6i 6i + 4 6i + 7 6m 6i + 6i + 6. Notice that this path starts at vertex 6i, ends at vertex 6i + 6, and uses one vertex from every other equivalence class modulo 6. It is therefore straightforward to verify that the concatenation of the paths Q 0, Q 6,..., Q 6m 6 is a hamiltonian cycle. Lemma.6 The digraph Circ(6m;,, 6m ) has a hamiltonian cycle. PROOF. The following is a hamiltonian cycle in this digraph: 0 6m 6 6m 4 6m 4 6m 5 1 6m 6m 0. The path from 0 to 6m 6, together with 6m (which immediately follows 6m 5) uses all of the vertices that are 0 (mod ); the path from 6m 4 to, together with 6m 1 (which immediately follows 6m ) uses all of the vertices that are (mod ), and the path from 4 to 6m 5, together with 1 and 6m, uses all of the vertices that are 1 (mod ). Although we do not use it in this paper, we recall the following elegant result that was mentioned in the introduction: Theorem.7 (R. A. Rankin, 1948, [, Thm. 4]) The circulant digraph Circ(n; a, b) of outdegree has a hamiltonian cycle iff there exist s, t Z +, such that s + t = gcd(n, a b), and gcd(n, sa + tb) = 1. Most cases of the proof In this section, we prove the following two results that cover most of the cases of Theorem 1.:

4 Proposition.1 If c > m, then Circ(6m;,, c) has a hamiltonian cycle. Proposition. If c m and c (mod 6), then Circ(6m;,, c) has a hamiltonian cycle. Notation. For convenience, let c = 6m c, so 1 c < 6m, and Circ(6m;,, c) = Circ(6m;,, c ). In fact, since the cases c = 6m, 6m and c 1 (mod n) have already been addressed, we may assume that c >. Remark.4 The use of c is very convenient when c is large (so one should think of c as being small less than m), but it can also be helpful in some other cases. Definition.5 A subdigraph P of Circ(n;,, c ) is a pseudopath from u to w if P is the disjoint union of a path from u to w and some number (perhaps 0) of cycles. In other words, if v is a vertex of P, then d + 0 if v = w; P (v) = 1 otherwise; and d 0 if v = u; P (v) = 1 otherwise. Definition.6 Let u, w be integers representing vertices of Circ(n;,, c). If u + c + w u + c, let P (u, w) be the pseudopath from u + 1 to w 1 whose vertex set is I(u, w), such that v travels by, if v I(u, w c ) I(u + c + 1, w ),, if v I(w c, u + c 1), c, if v {u + c, w}. Notice that the range of values for w makes sense because c >. Lemma.7 P (u, w) is a path if any of the following hold: w u c (mod ); or w u c + 1 (mod ) and w u c (mod ); or w u c + (mod ) and w u c (mod ). PROOF. To simplify the notation slightly, let us assume (without loss of generality) that u = 0. Case 1 Assume w c c (mod ). 4

5 Choose ε {1, }, such that w c ε 1 is even. The path in P (u, w) is 1 w c ε c c 0 c ε + w c w c c + ε w c + ε w 1. This contains both of the c-arcs, so there are no cycles in P (u, w). Case Assume there exists ε {1, }, such that w c ε c (mod ) and w c ε 1 is even. The path in P (u, w) is 1 w c ε c c 0 w c + ε w c w c c + ε c ε + w 1. This contains both of the c-arcs, so there are no cycles in P (u, w). Lemma.8 Let k Z be such that k 6m, c + k c +, and k + c (mod 6). Let u, w be integers representing vertices of Circ(n;,, c). Then for all u, w with u w and w u + 1 = k, the subgraph induced by I(u, w) has a hamiltonian path that starts at u + 1 and ends in {w 1, w}. PROOF. We consider three cases. Case 1 Assume k c + 1 (mod ). We have w u = k 1 c (mod ). Since k 1 {c + 1, c + }, we must have w u = k 1 c ). By Lemma.7, P (u, w) is a hamiltonian path from u + 1 to w 1. Case Assume k c + (mod ). Suppose, first, that k c + (so k c + 4). Letting w = w 1, then w u = w u 1 = k (c + 4) = c + and w u = (w 1) u = k c (mod ). By Lemma.7, P (u, w ) is a hamiltonian path in I(u, w ) from u + 1 to w 1. Adding the -arc from w 1 to w + 1 = w yields a hamiltonian path in I(u, w) from u + 1 to w. 5

6 Suppose instead that k = c +. Then w u = k 1 c + 1 (mod ) and w u = k 1 = (c + ) 1 c (mod ), so by Lemma.7, P (u, w) is a hamiltonian path from u + 1 to w 1. Case Assume k c (mod ). By assumption, we have k + c c + c 0 (mod ). Since k + c (mod 6), we must have k + c 0 (mod 6), so k c (mod ). Then w u = k 1 c + (mod ) and w u k 1 k c (mod ), so by Lemma.7, P (u, w) is a hamiltonian path from u + 1 to w 1. It is now easy to prove Propositions. and.1. PROOF OF PROPOSITION.. As previously mentioned, we may assume c >. Since < c m, we have m c < 6m, so c + < 6m < c +. Furthermore, since c (mod 6), we have 6m + c c (mod 6). Hence, Lemma.8 implies that the interval I(0, 6m 1) has a hamiltonian path from 1 to 6m or to 6m 1. Inserting the -edge from 6m to 1 or the -edge from 6m 1 to 1, yields a hamiltonian cycle. Since I(0, 6m 1) is the entire digraph, this completes the proof. PROOF OF PROPOSITION.1. We have already dealt with the cases c = 6m (in Lemma.5), c = 6m (in Lemma.6), and the cases 6m {c + 4, c + 6} are dealt with by Theorem 1.. Furthermore, we noted earlier that the case c = 6m 1 is clearly hamiltonian, so we may assume in what follows that c < 6m and 6m / {c + 4, c + 6}. Let K be the set of integers k that satisfy the conditions of Lemma.8. Note that c < m. We claim that n can be written as a sum n = k 1 + k + + k s, with each k i K. To see this, begin by noting that c +4 K (and 5 of any 6 consecutive integers between c + and c +, inclusive, belong to K). Thus, we may assume n < (c + 4) = c + 8, for otherwise it is easy to write n as a sum of integers in K. Since n is even, and n / {c + 4, c + 6}, we conclude that n = c + K. So n is obviously a sum of elements of K. This completes the proof of the claim. 6

7 The preceding paragraph implies that we may cover the vertices of Circ(n;,, c) by a disjoint collection of intervals I(u i, w i ), such that the number of vertices in I(u i, w i ) is k i. By listing the intervals in their natural order, we may assume u i+1 = w i + 1. By Proposition.8, the vertices of I(u i, w i ) can be covered by a path P i that starts at u i + 1 and ends in {w i 1, w i }. Since (u i+1 + 1) w i = (w i + ) w i = and (u i+1 + 1) (w i 1) = (w i + ) (w i 1) =, there is an arc from the terminal vertex of P i to the initial vertex of P i+1. Thus, by adding a number of -arcs and/or -arcs, we may join all of the paths P 1, P,..., P s into a single cycle that covers all of the vertices of Circ(n;,, c). Thus, we have constructed a hamiltonian cycle. 4 The remaining cases In this section, we prove the following result. Combining it with Propostions.1 and. (and Theorem 1.) completes the proof of Theorem 1.. Proposition 4.1 If c m and c (mod 6), then Circ(6m;,, c) has a hamiltonian cycle. Definition 4. Let t be any natural number, such that 0 6t c 9. (1) Let l 1 = c 5, l = l (t) = c 1 + 6t, l = c, l 4 = c +. () Define subdigraphs Q 1, Q, Q and Q 4 of Circ(6m;,, c) as follows: The vertex set of Q i is I(0, l i + ) {l i + 5}. c, if v = 0; In Q 1, vertex v travels by, if v = 1 or ;, if v =, 4,..., c 6. c, if v = 1 or 6t + 4; In Q, vertex v travels by, if v = or 6t + 5 v c ;, if v = 0 or v 6t + or c 1 v c + 6t. c, if v = 1 or ; In Q, vertex v travels by, if v = 1, or 4 v c. 7

8 c, if v = or 8; In Q 4, vertex v travels by, if 9 v c 1;, if v = 0, 1 or v 7 or c v c +. Notation 4. For ease of later referral, we also let l i (t) denote l i for i {1,, 4}. Lemma 4.4 (1) Each Q i is the union of four disjoint paths from {0, 1,, 5} to {l i, l i + 1, l i +, l i + 5}. () Indeed, if we let u 1 = 0, u = 1, u =, u 4 = 5, and w i,j = l i + u j, and define permutations σ 1 = (14), σ = (4), σ = (14), and σ 4 = identity, then Q i contains a path from u j to w i,σi (j) for j = 1,,, 4. PROOF. The paths in Q 1 are: 0 c c (= l 1 + 5), 1 c (= l 1 + ), 4 c 5 (= l 1 ), 5 c 4 (= l 1 + 1). The paths in Q are: 0 6t + 6 c 1 c 1 + 6t (= l ), 1 c c + 1 c t (= l + ), 4 6t + 4 c c t (= l + 5), 5 6t + 5 c c + 6t (= l + 1). The paths in Q are: 0 c c (= l + ), 1 c c + (= l + 5), c 1 (= l + 1), 5 c (= l ). 8

9 The paths in Q 4 are: 0 9 c c + (= l4 ), 1 10 c + 1 c + 4 (= l 4 + 1), c c + c + 5 (= l 4 + ), 5 8 c c + 8 (= l 4 + 5). The above lemma yields the following conclusion: Lemma 4.5 If, for some natural number k, there exist sequences I = (i 1, i,..., i k ) with each i j {1,,, 4}, and T = (t 1, t,..., t k ) with 0 6t j c 9, for each j, such that (i) k j=1 l ij (t j ) = 6m, and (ii) the permutation product σ i1 σ i σ ik is a cycle of length 4, then Circ(6m;,, c) has a hamiltonian cycle constructed by concatenating copies of Q 1, Q, Q, and Q 4. PROOF OF PROPOSITION 4.1. Since σ 4 is the identity and l 4 = c +, we see that if Circ(6m;,, c) has a hamiltonian cycle constructed by concatenating copies of Q 1, Q, Q, and Q 4, then Circ(6m + c + ;,, c) also has such a hamiltonian cycle. Thus, by subtracting some multiple of c + from 6m, we may assume c 6 6m c 9. (For this modified c, it is possible that c > m.) Recall that 0 6t c 9, so c 6 + 6t can be any multiple of 6 between c 6 and c 15. Since σ 1 σ = (14) and l 1 + l (t) = c 6 + 6t, it follows that Circ(6m;,, c) has a hamiltonian cycle constructed by concatenating one copy of Q 1 and one copy of Q whenever c 6 6m c 15. The only case that remains is when 6m = c 9. Now σ 1 σ = (14) and l 1 + l = c 9, so Circ(c 9;,, c) has a hamiltonian cycle constructed by concatenating one copy of Q 1 and two copies of Q. 9

10 References [1] Stephen C. Locke and Dave Witte: On non-hamiltonian circulant digraphs of outdegree three, J. Graph Theory 0 (1999), no. 4, MR [] R. A. Rankin: A campanological problem in group theory, Proc. Cambridge Philos. Soc. 44 (1948) MR