1. Groups with elements of order 8 do not have the DCI propertyTed Dobson, Joy Morris, Pablo Spiga, 2025, original scientific article Abstract: Let k be odd, and n an odd multiple of 3. Although this can also be deduced from known results, we provide a new proof that Ck ⋊ C₈ and (Cn × C₃) ⋊ C₈ do not have the Directed Cayley Isomorphism (DCI) property. When k is prime, Ck ⋊ C₈ had previously been proved to have the Cayley Isomorphism (CI) property. To the best of our knowledge, the groups Cp ⋊ C₈ (where p is an odd prime) are only the second known infinite family of groups that have the CI property but do not have the DCI property. This also provides a new proof of the result (which follows from known results but was not explicitly published) that no group with an element of order 8 has the DCI property.
One piece of our proof is a new result that may prove to be of independent interest: we show that if a permutation group has a regular subgroup of index 2 then it must be 2-closed.
Keywords: CI property, DCI property, Cayley graphs, Cayley digraphs, 2-closed groups, 2-closure
Published in RUP: 03.11.2025; Views: 760; Downloads: 5
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2. Colour-permuting automorphisms of complete Cayley graphsShirin Alimirzaei, Dave Witte Morris, 2025, original scientific article Abstract: Let G be a (finite or infinite) group, and let KG = Cay(G; G \ {1}) be the complete graph with vertex set G, considered as a Cayley graph of G. Being a Cayley graph, it has a natural edge-colouring by sets of the form {s, s-1} for s in G. We prove that every colour-permuting automorphism of KG is an affine map, unless G is isomoprhic to the direct product of Q8 and B, where Q8 is the quaternion group of order 8, and B is an abelian group, such that b2 is trivial for all b in B.
We also prove (without any restriction on G) that every colour-permuting automorphism of KG is the composition of a group automorphism and a colour-preserving graph automorphism. This was conjectured by D. P. Byrne, M. J. Donner, and T. Q. Sibley in 2013.
Keywords: Cayley graph, automorphism, colour-permuting, complete graphs
Published in RUP: 03.11.2025; Views: 744; Downloads: 8
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3. On regular graphs with Šoltés verticesNino Bašić, Martin Knor, Riste Škrekovski, 2025, original scientific article Abstract: Let ▫$W(G)$▫ be the Wiener index of a graph ▫$G$▫. We say that a vertex ▫$v \in V(G)$▫ is a Šoltés vertex in ▫$G$▫ if ▫$W(G - v) = W(G)$▫, i.e. the Wiener index does not change if the vertex ▫$v$▫ is removed. In 1991, Šoltés posed the problem of identifying all connected graphs ▫$G$▫ with the property that all vertices of ▫$G$▫ are Šoltés vertices. The only such graph known to this day is ▫$C_{11}$▫. As the original problem appears to be too challenging, several relaxations were studied: one may look for graphs with at least ▫$k$▫ Šoltés vertices; or one may look for ▫$\alpha$▫-Šoltés graphs, i.e. graphs where the ratio between the number of Šoltés vertices and the order of the graph is at least ▫$\alpha$▫. Note that the original problem is, in fact, to find all ▫$1$▫-Šoltés graphs. We intuitively believe that every ▫$1$▫-Šoltés graph has to be regular and has to possess a high degree of symmetry. Therefore, we are interested in regular graphs that contain one or more Šoltés vertices. In this paper, we present several partial results. For every ▫$r\ge 1$▫ we describe a construction of an infinite family of cubic ▫$2$▫-connected graphs with at least ▫$2^r$▫ Šoltés vertices. Moreover, we report that a computer search on publicly available collections of vertex-transitive graphs did not reveal any ▫$1$▫-Šoltés graph. We are only able to provide examples of large ▫$\frac{1}{3}$▫-Šoltés graphs that are obtained by truncating certain cubic vertex-transitive graphs. This leads us to believe that no ▫$1$▫-Šoltés graph other than ▫$C_{11}$▫ exists.
Keywords: Šoltés problem, Wiener index, regular graphs, cubic graphs, Cayley graph, Šoltés vertex
Published in RUP: 10.09.2025; Views: 1112; Downloads: 9
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4. A novel characterization of cubic Hamiltonian graphs via the associated quartic graphsSimona Bonvicini, Tomaž Pisanski, 2017, original scientific article Abstract: We give a necessary and sufficient condition for a cubic graph to be Hamiltonian by analyzing Eulerian tours in certain spanning subgraphs of the quartic graph associated with the cubic graph by 1-factor contraction. This correspondence is most useful in the case when it induces a blue and red 2-factorization of the associated quartic graph. We use this condition to characterize the Hamiltonian ▫$I$▫-graphs, a further generalization of generalized Petersen graphs. The characterization of Hamiltonian ▫$I$▫-graphs follows from the fact that one can choose a 1-factor in any ▫$I$▫-graph in such a way that the corresponding associated quartic graph is a graph bundle having a cycle graph as base graph and a fiber and the fundamental factorization of graph bundles playing the role of blue and red factorization. The techniques that we develop allow us to represent Cayley multigraphs of degree 4, that are associated to abelian groups, as graph bundles. Moreover, we can find a family of connected cubic (multi)graphs that contains the family of connected ▫$I$▫-graphs as a subfamily.
Keywords: generalized Petersen graphs, I-graphs, Hamiltonian cycles, Eulerian tours, Cayley multigraphs
Published in RUP: 03.01.2022; Views: 2960; Downloads: 30
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8. Hamiltonian cycles in Cayley graphs whose order has few prime factorsKlavdija Kutnar, Dragan Marušič, D. W. Morris, Joy Morris, Primož Šparl, 2012, original scientific article Abstract: 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 ▫$24 \ne k < 32$▫, or of the form ▫$kpq$▫ with ▫$k \le 5$▫, or of the form ▫$pqr$▫, or of the form ▫$kp^2$▫ with ▫$k \le 4$▫, or of the form ▫$kp^3$▫ with ▫$k \le 2$▫.
Keywords: graph theory, Cayley graphs, hamiltonian cycles
Published in RUP: 15.10.2013; Views: 6620; Downloads: 132
Full text (545,91 KB) |
