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Povzetek: We describe all the Bier spheres of dimension $d$ with chromatic number equal to $d+1$ and prove that all other $d$-dimensional Bier spheres have chromatic number equal to $d+2$, for any integer $d \ge 0$. Then we prove a general formula for complex and mod $p$ Buchstaber numbers of a Bier sphere ${\rm Bier}(K)$, for each prime $p \in {\mathbb N}$ in terms of the $f$-vector of the underlying simplicial complex $K$. Finally, we classify all chordal Bier spheres and obtain their canonical realizations as boundaries of stacked polytopes.
Ključne besede: Bier sphere, Buchstaber number, chordal graph, chromatic number, stacked polytope
Objavljeno v DiRROS: 06.05.2026; Ogledov: 64; Prenosov: 39
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Povzetek: Given a graph $G$, a subset $M\subseteq V(G)$ is a mutual-visibility (MV) set if for every $u,v\in M$, there exists a $u,v$-geodesic whose internal vertices are not in $M$. We investigate proper vertex colorings of graphs whose color classes are mutual-visibility sets. The main concepts that arise in this investigation are independent mutual-visibility (IMV) sets and vertex partitions into these sets (IMV colorings). The IMV number $\mu_{i}$ and the IMV chromatic number $\chi_{\mu_{i}}$ are defined as maximum and minimum cardinality taken over all IMV sets and IMV colorings, respectively. Along the way, we also continue with the study of MV chromatic number $\chi_{\mu}$ (as the smallest number of sets in a vertex partition into MV sets), which was initiated in an earlier paper. We establish a close connection between the (I)MV chromatic numbers of subdivisions of complete graphs and Ramsey numbers $R(4^k;2)$. From the computational point of view, we prove that the problems of computing $\chi_{\mu_{i}}$ and $\mu_{i}$ are NP-complete, and that it is NP-hard to decide whether a graph $G$ satisfies $\mu_i(G)=\alpha(G)$ where $\alpha(G)$ is the independence number of $G$. Several tight bounds on $\chi_{\mu_{i}}$, $\chi_{\mu}$ and $\mu_{i}$ are given. Exact values/formulas for these parameters in some classical families of graphs are proved. In particular, we prove that $\chi_{\mu_{i}}(T)=\chi_{\mu}(T)$ holds for any tree $T$ of order at least $3$, and determine their exact formulas in the case of lexicographic product graphs. Finally, we give tight bounds on the (I)MV chromatic numbers for the Cartesian and strong product graphs, which lead to exact values in some important families of product graphs.
Ključne besede: independent mutual visibility, mutual-visibility coloring, independence number, Ramsey number, graph product, diameter 2 graph, geodesic
Objavljeno v DiRROS: 05.05.2026; Ogledov: 116; Prenosov: 73
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Povzetek: A predominated graph is a pair $(G,D)$, where $G$ is a graph and the vertices in $D\subseteq V(G)$ are considered already dominated. Maker-Breaker domination game critical (MBD critical) predominated graphs are introduced as the predominated graphs $(G,D)$ on which Staller wins the game, but Dominator wins on $(G, D \cup \{v\})$ for every vertex $v \in V(G) \setminus D$. Tools are developed for handling the Maker-Breaker domination game on trees which lead to a characterization of Staller-win predominated trees. MBD critical predominated trees are characterized and an algorithm is designed which verifies in linear time whether a given predominated tree is MBD critical. A large class of MBD critical predominated cacti is presented and Maker-Breaker critical hypergraphs are constructed.
Ključne besede: domination games, Maker-Breaker games, Maker-Breaker domination game, predomination, hypergraph
Objavljeno v DiRROS: 16.04.2026; Ogledov: 127; Prenosov: 78
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Povzetek: We say that a ring is strongly (resp. weakly) left Jacobson if every semiprime (resp. prime) left ideal is an intersection of maximal left ideals. There exist Jacobson rings that are not weakly left Jacobson, e.g. the Weyl algebra. Our main result is the following one-sided noncommutative Nullstellensatz: For any finite-dimensional ${\mathbb F}$-algebra ${\mathbb A}$ the ring ${\mathbb A}[x_1, \ldots,x_n]$ of polynomials with coefficients in ${\mathbb A}$ is strongly left Jacobson and every maximal left ideal of ${\mathbb A}[x_1, \ldots,x_n]$ has finite codimension. We also prove that an Azumaya algebra is strongly left Jacobson iff its center is Jacobson and that an algebra that is a finitely generated module over its center is weakly left Jacobson iff it is Jacobson.
Ključne besede: Nullstellensatz, noncommutative geometry, maximal left ideals, Jacobson ring, Azumaya algebra, Weyl algebra
Objavljeno v DiRROS: 14.04.2026; Ogledov: 143; Prenosov: 101
Celotno besedilo (1,02 MB)
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Povzetek: This paper presents a comprehensive and constructive analysis of degree 7 double Pythagorean-hodograph (DPH) curves for the three distinct structural classes. A unified framework is introduced for their construction, particularly useful for interpolation tasks involving prescribed boundary data. The approach explicitly identifies the degrees of freedom available in each class and distinguishes between helical and non-helical curve types. Furthermore, the structure of a specific rational curve in the complex plane that via the normalized Hopf map generates the tangent indicatrix is revealed for all degree 7 DPH curves. This confirms known results for helical curves and extends the interpretation to non-helical cases. The practical applicability of the derived curves is demonstrated through a numerical interpolation example, which also validates the stated number of degrees of freedom.
Ključne besede: double Pythagorean-hodograph curves, helical/non-helical curves, quaternions, Hopf map, Frenet frame, interpolation
Objavljeno v DiRROS: 14.04.2026; Ogledov: 107; Prenosov: 91
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