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Abstract: A $\Gamma$-distance magic labeling of a graph $G = (V, E)$ with $|V| = n$ is a bijection $\ell$ from $V$ to an Abelian group $\Gamma$ of order $n$, for which there exists $\mu \in \Gamma$, such that the weight $w(x) =\sum_{y\in N(x)}\ell(y)$ of every vertex $x \in V$ is equal to $\mu$. In this case, the element $\mu$ is called the magic constant of $G$. A graph $G$ is called a group distance magic if there exists a $\Gamma$-distance magic labeling of $G$ for every Abelian group $\Gamma$ of order $n$. In this paper, we focused on cubic $\Gamma$-distance magic graphs as well as some properties of such graphs.
Keywords: group distance magic labeling, Kotzig array, generalized Petersen graph
Published in RUP: 06.05.2026; Views: 320; Downloads: 6
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Abstract: We present some methods for constructing connected spatial geometric configurations (p_q, n_k) of points and lines, preserved by the same isometries of Euclidean space E³ as the predetermined Platonic solid. In this paper, we are mainly interested in configurations (n₃), (n₄), and (n₅), but also in unbalanced configurations (p₃, n₄), (p₃, n₅), and (p₄, n₅).
Keywords: configuration of points and lines, symmetry group, Platonic solid, centrally symmetric solid, projection from a point
Published in RUP: 22.12.2025; Views: 470; Downloads: 1
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Abstract: In 1965, Erdős and Pósa proved that there is an (approximate) duality between the maximum size of a packing of cycles and the minimum size of a vertex set hitting all cycles. Such a duality does not hold for odd cycles, and Dejter and Neumann-Lara asked in 1988 to find all pairs (l, z) of integers where such a duality holds for the family of cycles of length l modulo z. We characterise all such pairs, and we further generalise this characterisation to cycles in graphs labelled with a bounded number of abelian groups, whose values avoid a bounded number of elements of each group. This unifies almost all known types of cycles that admit such a duality, and it also provides new results. Moreover, we characterise the obstructions to such a duality in this setting, and thereby obtain an analogous characterisation for cycles in graphs embeddable on a fixed compact orientable surface.
Keywords: Erdős-Pósa property, cycle packing, group-labelled graph
Published in RUP: 17.11.2025; Views: 748; Downloads: 9
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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: 590; Downloads: 4
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Abstract: Determining an upper bound on s for finite vertex-primitive s-arc-transitive digraphs has received considerable attention dating back to a question of Praeger in 1990. It was shown by Giudici and Xia that the smallest upper bound on s is attained for some digraph admitting an almost simple s-arc-transitive group. In this paper, based on the work of Pan, Wu and Yin, we prove that s<=2 in the case where the group is an alternating or symmetric group.
Keywords: digraph, vertex-primitive, s-arc-transitive, alternating group, symmetric group
Published in RUP: 22.10.2025; Views: 641; Downloads: 6
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