Authors:Paola Bonacini, Selda Küçükçifçi, Salvatore Milici, E. Şule Yazıcı Pages: 1 - 22 Abstract: In this paper we consider the uniformly resolvable decompositions of the complete graph $K_v$ minus a 1-factor $(K_v − I)$ into subgraphs where each resolution class contains only blocks isomorphic to the same graph. We completely determine the spectrum for the case in which all the resolution classes consist of either 4-cycles or 3-stars. PubDate: 2022-12-29 Issue No:Vol. 17, No. 2 (2022)
Authors:Miklós Eper, Jenő Szirmai Pages: 23 - 40 Abstract: After the investigation of the congruent and non-congruent hyperball packings related to doubly truncated Coxeter orthoscheme tilings [Acta Univ. Sapientiae, Mathematica, 11, 2 (2019), 437–459], we consider the corresponding covering problems. In Non-fundamental trunc-simplex tilings and their optimal hyperball packings and coverings in hyperbolic space the authors gave a partial classification of supergroups of some hyperbolic space groups whose fundamental domains will be integer parts of truncated tetrahedra, and determined the optimal congruent hyperball packing and covering configurations belonging to some of these classes. In this paper, we complement these results with the investigation of the non-congruent covering cases and the remaining congruent cases. We prove, that between congruent and non-congruent hyperball coverings the thinnest belongs to the $\{7,3,7\}$ Coxeter tiling with density $\approx 1.26829$. This covering density is smaller than the conjectured lower bound density of L. Fejes Tóth for coverings with balls and horoballs. We also study the local packing arrangements related to $\{u,3,7\}$ $(6< u < 7, ~ u\in \mathbb{R})$ doubly truncated orthoschemes and the corresponding hyperball coverings. We prove, that these coverings are achieved their minimum density at parameter $u\approx 6.45953$ with covering density $\approx 1.26454$ which is smaller than the above record-small density, but this hyperball arrangement related to this locally optimal covering can not be extended to the entire $\mathbb{H}^3$. Moreover, we see that in the hyperbolic plane $\mathbb{H}^2$ the universal lower bound of the congruent circle, horocycle, hypercycle covering density $\sqrt{12}/\pi$ can be approximated arbitrarily well also with non-congruent hypercycle coverings generated by doubly truncated Coxeter orthoschemes. PubDate: 2022-12-29 Issue No:Vol. 17, No. 2 (2022)
Authors:David Richter Pages: 41 - 66 Abstract: A rectangulation is a subdivision of a rectangle into rectangles. A generic rectangulation is a rectangulation that has no crossing segments. We explain several observations and pose some questions about generic rectangulations. In particular, we show how one may "centrally invert" a generic rectangulation about any given rectangle, analogous to reflection across a circle in classical geometry. We also explore 3-dimensional orthogonal polytopes related to "marked" rectangulations and drawings of planar maps. These observations arise from viewing a generic rectangulation as topologically equivalent to a sphere. PubDate: 2022-12-29 Issue No:Vol. 17, No. 2 (2022)
Authors:Jacob Anthony White Pages: 67 - 76 Abstract: The two-fold chromatic number of a graph is the minimum number of colors needed to ensure that there is a way to color the graph so that each vertex gets two distinct colors, and adjacent vertices have no colors in common. The Ore degree is the maximum sum of degrees of an edge in a graph. We prove that, for 2-connected graphs, the two-fold chromatic number is at most the Ore degree, unless G is a complete graph or an odd cycle. PubDate: 2022-12-29 Issue No:Vol. 17, No. 2 (2022)
Authors:Zia Ullah Khan, Xiao-Dong Zhang Pages: 77 - 95 Abstract: Let $G$ be a connected graph with $\alpha \in [0,1]$, the $D_{\alpha}$-spectral radius of $G$ is defined to be the spectral radius of the matrix $D_{\alpha}(G)$, defined as $D_{\alpha}(G)= \alpha T(G)+(1 - \alpha)D(G)$, where $T(G)$ is a transmission diagonal matrix of $G$ and $D(G)$ denotes the distance matrix of $G$. In this paper, we give some sharp upper and lower bounds for the $D_{\alpha}$-spectral radius with respect to different graph parameters. PubDate: 2022-12-29 Issue No:Vol. 17, No. 2 (2022)
Authors:Abdulaziz Alanazi, Bashair Alenazi, William Keith, Augustine Munagi Pages: 96 - 111 Abstract: We study new classes of overpartitions of numbers based on the properties of non-overlined parts. Several combinatorial identities are established by means of generating functions and bijective proofs. We show that our enumeration function satisfies a pair of infinite Ramanujantype congruences modulo 3. Lastly, by conditioning on the overlined parts of overpartitions, we give a seemingly new identity between the number of overpartitions and a certain class of ordinary partition functions. A bijective proof for this theorem also includes a partial answer to a previous request for a bijection on partitions doubly restricted by divisibility and frequency. PubDate: 2022-12-29 Issue No:Vol. 17, No. 2 (2022)
Authors:Anthony Forbes, Terry Griggs Pages: 112 - 136 Abstract: The design spectrum has been determined for ten of the 15 graphs with six vertices and ten edges. In this paper, we solve the design spectrum problem for the remaining five graphs with three possible exceptions. PubDate: 2022-12-29 Issue No:Vol. 17, No. 2 (2022)
Authors:Arthur Busch, Mohammed Mutar, Daniel Slilaty Pages: 137 - 149 Abstract: Zaslavsky observed that the topics of directed cycles in directed graphs and alternating cycles in edge 2-colored graphs have a common generalization in the study of coherent cycles in bidirected graphs. There are classical theorems by Camion, Harary and Moser, Häggkvist and Manoussakis, and Saad which relate strong connectivity and Hamiltonicity in directed "complete" graphs and edge 2-colored "complete" graphs. We prove two analogues to these theorems for bidirected "complete" signed graphs. PubDate: 2022-12-29 Issue No:Vol. 17, No. 2 (2022)
Authors:Craig Kaplan Pages: 150 - 171 Abstract: A shape's Heesch number is the number of layers of copies of the shape that can be placed around it without gaps or overlaps. Experimentation and exhaustive searching have turned up examples of shapes with finite Heesch numbers up to six, but nothing higher. The computational problem of classifying simple families of shapes by Heesch number can provide more experimental data to fuel our understanding of this topic. I present a technique for computing Heesch numbers of nontiling polyforms using a SAT solver, and the results of exhaustive computation of Heesch numbers up to 19-ominoes, 17-hexes, and 24-iamonds. PubDate: 2022-12-29 Issue No:Vol. 17, No. 2 (2022)
Authors:Abderrahim Boussaïri, Soufiane Lakhlifi, Imane Talbaoui Pages: 172 - 186 Abstract: An $n$-tournament $T$ with vertex set $V$ is simple if there is no subset $M$ of $V$ such that $2\leq \left \vert M\right \vert \leq n-1$ and for every $x\in V\setminus M$, either $M\rightarrow x$ or $x\rightarrow M$. The simplicity index of an $n$-tournament $T$ is the minimum number $s(T)$ of arcs whose reversal yields a nonsimple tournament. Müller and Pelant (1974) proved that $s(T)\leq(n-1)/2$, and that equality holds if and only if $T$ is doubly regular. As doubly regular tournaments exist only if $n\equiv 3\pmod{4}$, $s(T)<(n-1)/2$ for $n\not\equiv3\pmod{4}$. In this paper, we study the class of $n$-tournaments with maximal simplicity index for $n\not\equiv3\pmod{4}$. PubDate: 2022-12-29 Issue No:Vol. 17, No. 2 (2022)
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