Faculty Research and Publications

Permanent URI for this community

https://opus.uleth.ca/handle/10133/4940

Browse

Browsing Faculty Research and Publications by Author "Kutnar, Klavdija"

Now showing 1 - 3 of 3
  • Thumbnail Image
    Item
    Hamiltonian cycles in Caley graphs whose order has few prime factors
    (Drustvo Matematikov, Fizikov in Astronomov, 2012) Kutnar, Klavdija; Marusic, Dragan; Morris, Dave Witte; Morris, Joy; Sparl, Primoz
    We prove that if Cay(G;S) is a connected Cayley graph with n vertices,and the prime factorization of n is very small, then Cay(G;S) has a hamiltonian cycle. More precisely, if p, q, and r are distinct primes, then n can be of the form kp with 24 6= k < 32, or of the form kpq with k ≤ 5,or of the form pqr,or of the form kp2 with k ≤ 4,or of the form kp3 with k ≤ 2.
  • Thumbnail Image
    Item
    On colour-preserving automorphisms of Caley graphs
    (Drustvo Matematikov, Fizikov in Astronomov, 2016) Hujdurovic, Ademir; Kutnar, Klavdija; Morris, Dave Witte; Morris, Joy
    We study the automorphisms of a Cayley graph that preserve its natural edge-colouring. More precisely, we are interested in groups G, such that every such automorphism of every connected Cayley graph on G has a very simple form: the composition of a left-translation and a group automorphism. We find classes of groups that have the property, and we determine the orders of all groups that do not have the property. We also have analogous results for automorphisms that permute the colours, rather than preserving them.
  • Thumbnail Image
    Item
    Vertex-transitive digraphs with extra automorphisms that preserve the natural arc-colouring
    (The University of Queensland, Centre for Discrete Mathematics and Computing, 2017) Dobson, Ted; Hujdurovic, Ademir; Kutnar, Klavdija; Morris, Joy
    In a Cayley digraph on a group G, if a distinct colour is assigned to each arc-orbit under the left-regular action of G, it is not hard to show that the elements of the left-regular action of G are the only digraph automorphisms that preserve this colouring. In this paper, we show that the equivalent statement is not true in the most straightforward generalisation to G-vertex-transitive digraphs, even if we restrict the situation to avoid some obvious potential problems. Specifically, we display an infinite family of 2-closed groups G, and a G-arc-transitive digraph on each (without any digons) for which there exists an automorphism of the digraph that is not an element of G (it is an automorphism of G). Since the digraph is G-arc-transitive, the arcs would all be assigned the same colour under the colouring by arc-orbits, so this digraph automorphism is colour-preserving.