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37. Grid-based graphs, linear realizations, and the Buratti-Horak-Rosa conjectureM. A. Ollis, Onur Ağirseven, 2026, original scientific article Abstract: Label the vertices of the complete graph K_v with the integers {0, 1, …, v − 1} and define the length l of the edge between distinct vertices labeled x and y by ℓ(x, y) = min (|y − x|,v − |y − x|). A realization of a multiset L of size v − 1 is a Hamiltonian path through K_v whose edge labels are L. The Buratti-Horak-Rosa (BHR) Conjecture is that there is a realization for a multiset L if and only if for any divisor d of v the number of multiples of d in L is at most v − d.
We introduce “grid-based graphs" as a useful tool for constructing particular types of realizations, called “linear realizations,"especially when the multiset in question has a support of size 3. This lets us prove many new instances of the BHR Conjecture, including those for multisets of the form {1^a, x^b, y^c} when a ≥ x + y − ϵ, where ϵ is the number of even elements in {x, y}, and those for all multisets of the following forms for sufficiently large v with gcd (v,y) = 1 for all y ∈ L:
•{1^a, 2^b, x^c}, except possibly when a ∈ {1, 2} and x is odd,
•{1^a, x^b, (x + 1)^c}.
This establishes that there are infinitely many sets U of size 3 for which there are infinitely many values of v where the BHR Conjecture holds for each multiset with support U. We also show that the BHR Conjecture holds for {1^a, x^b, (x + 1)^c} when x ∈ {7, 9, 10} and gcd (v, x) = gcd (v, x + 1) = 1.
Keywords: complete graph, Hamiltonian path, edge-length, realization, grid-based graph
Published in RUP: 28.09.2026; Views: 252; Downloads: 3
 Full text (509,79 KB) |
38. Colouring a graph with position setsUllas Chandran S.V., Gabriele Di Stefano, Haritha S., Elias John Thomas, James Tuite, 2026, original scientific article Abstract: In this paper we consider a colouring version of the general position problem. The gp-chromatic number is the smallest number of colours needed to colour V(G) such that each colour class has the no-three-in-line property. We determine bounds on this colouring number in terms of the diameter, general position number, size, chromatic number, cochromatic number and total domination number and prove realisation results. We also determine it for several graph classes, including Kneser graphs K(n,2), line graphs of complete graphs, complete multipartite graphs, block graphs and Cartesian products. We also show that the gp-colouring problem is NP-complete.
Keywords: general position set, monophonic position set, graph colouring, Cartesian product, complexity
Published in RUP: 28.09.2026; Views: 196; Downloads: 2
Full text (486,44 KB) |
39. On a conjecture by Hayashi on finite connected quandlesAntónio Lages, Pedro Lopes, 2026, original scientific article Abstract: A quandle is an algebraic structure whose binary operation is idempotent, right-invertible and right self-distributive. Right-invertibility ensures right translations are permutations and right self-distributivity ensures further they are automorphisms. For finite connected quandles, all right translations have the same cycle structure, called the profile of the connected quandle. Hayashi conjectured that the longest length in the profile of a finite connected quandle is a multiple of the remaining lengths. We prove that this conjecture is true for profiles with at most five lengths.
Keywords: quandles, right translations, cycles, profiles, Hayashi's conjecture
Published in RUP: 28.09.2026; Views: 178; Downloads: 3
Full text (378,93 KB) |
40. A skew group ring of Z/2Z over U(sl_2), Leonard triples and odd graphsHau-Wen Huang, Chin-Yen Lee, 2026, original scientific article Abstract: We employ a skew group ring of Z/2Z over U(sl₂) to construct modules over the universal Bannai–Ito algebra. Furthermore, we characterize the conditions under which the defining generators act as Leonard triples on these modules. As a combinatorial realization, we establish a surjective algebra homomorphism from the universal Bannai–Ito algebra to the Terwilliger algebra of an odd graph. This homomorphism provides a unified description of Leonard triples on all irreducible modules over the Terwilliger algebra.
Keywords: Leonard triples, odd graphs, skew group rings, Terwilliger algebras
Published in RUP: 28.09.2026; Views: 226; Downloads: 2
Full text (588,49 KB) |
