Discrete Mathematics 309 (2009) 5484–5490
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Discrete Mathematics
journal homepage: www.elsevier.com/locate/disc
Hamiltonian cycles in (2, 3, c)-circulant digraphs
Dave Witte Morris
∗
, Joy Morris, Kerri Webb
Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Alberta, T1K 3M4, Canada
a r t i c l e i n f o
Article history:
Received 30 December 2008
Accepted 6 January 2009
Available online 3 February 2009
Keywords:
Hamiltonian cycle
Circulant
Directed graph
a b s t r a c t
Let D be the circulant digraph with n vertices and connection set {2, 3, c}. (Assume D is
loopless and has outdegree 3.) Work of S. C. Locke and D. Witte implies that if n is a multiple
of 6, c ∈ {(n/2) + 2, (n/2) + 3}, and c is even, then D does not have a hamiltonian cycle.
For all other cases, we construct a hamiltonian cycle in D.
© 2009 Elsevier B.V. All rights reserved.
1. Introduction
For S ⊂ Z
n
, the circulant digraph with vertex set Z
n
and arcs from v to v + s for each v ∈ Z
n
and s ∈ S is denoted by
Circ(n; S). A fundamental open problem is to determine which circulant digraphs have hamiltonian cycles. By the following
elegant result, circulant digraphs of outdegree three are the smallest digraphs that need to be considered.
Theorem 1.1 (Rankin, 1948 [2, Thm. 4]). The circulant digraph Circ(n; a, b) of outdegree 2 has a hamiltonian cycle iff there exist
s, t ∈ Z
≥0
, such that:
• s + t = gcd(n, a − b), and
• gcd(n, sa + tb) = 1.
Locke and Witte [1] found two infinite families of non-hamiltonian circulant digraphs of outdegree 3; one of the families
includes the following examples.
Theorem 1.2 (Locke-Witte, cf. [1, Thm. 1.4]).
(1) Circ(6m; 2, 3, 3m + 2) is not hamiltonian if and only if m is even.
(2) Circ(6m; 2, 3, 3m + 3) is not hamiltonian if and only if m is odd.
In this paper, we show that the above examples are the only loopless digraphs of the form Circ(n; 2, 3, c) that have
outdegree 3 and are not hamiltonian:
Theorem 1.3. Assume c ≡ 0, 2, 3 (mod n). The digraph Circ(n; 2, 3, c) is not hamiltonian iff all of the following hold:
(1) n is a multiple of 6 , so we may write n = 6m,
(2) either c ≡ 3m + 2 (mod n) or c ≡ 3m + 3 (mod n), and
(3) c is even.
The direction (⇐) of Theorem 1.3 is a restatement of part of the Locke-Witte Theorem (1.2), so we need only prove the
opposite direction.
∗
Corresponding author.
E-mail addresses: Dave.Morris@uleth.ca (D.W. Morris), Joy.Morris@uleth.ca (J. Morris), Kerri.Webb@uleth.ca (K. Webb).
0012-365X/$ – see front matter © 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.disc.2009.01.001