Borsuk and V\'azsonyi problems through Reuleaux polyhedra

The Borsuk conjecture and the V\'azsonyi problem are two attractive and famous questions in discrete and combinatorial geometry, both based on the notion of diameter of a bounded sets. In this paper, we present an equivalence between the critical sets with Borsuk number 4 in $\mathbb{R}^3$ and the minimal structures for the V\'azsonyi problem by using the well-known Reuleaux polyhedra. The latter lead to a full characterization of all finite sets in $\mathbb{R}^3$ with Borsuk number 4. The proof of such equivalence needs various ingredients, in particular, we proved a conjecture dealing with strongly critical configuration for the V\'azsonyi problem and showed that the diameter graph arising from involutive polyhedra is vertex (and edge) 4-critical

Nodal sets of magnetic Schroedinger operators of Aharonov-Bohm type and energy minimizing partitions

In this paper we consider a stationary Schroedinger operator in the plane, in presence of a magnetic field of Aharonov-Bohm type with semi-integer circulation. We analyze the nodal regions for a class of solutions such that the nodal set consists of regular arcs, connecting the singular points with the boundary. In case of one magnetic pole, which is free to move in the domain, the nodal lines may cluster dissecting the domain in three parts. Our main result states that the magnetic energy is critical (with respect to the magnetic pole) if and only if such a configuration occurs. Moreover the nodal regions form a minimal 3-partition of the domain (with respect to the real energy associated to the equation), the configuration is unique and depends continuously on the data. The analysis performed is related to the notion of spectral minimal partition introduced in [20]. As it concerns eigenfunctions, we similarly show that critical points of the Rayleigh quotient correspond to multiple clustering of the nodal lines.Comment: 32 page

Projective freeness and Hermiteness of complex function algebras

The paper studies projective freeness and Hermiteness of algebras of complex-valued continuous functions on topological spaces, Stein algebras, and commutative unital Banach algebras. New sufficient cohomology conditions on the maximal ideal spaces of the algebras are given that guarantee the fulfilment of these properties. The results are illustrated by nontrivial examples. Based on the Borsuk theory of shapes, a new class $\mathscr{C}$ of commutative unital complex Banach algebras is introduced (an analog of the class of local rings in commutative algebra) such that the projective tensor product with algebras in $\mathscr C$ preserves projective freeness and Hermiteness. Some examples of algebras of class $\mathscr{C}$ and of other projective free and Hermite function algebras are assembled. These include, e.g., Douglas algebras, finitely generated algebras of symmetric functions, Bohr-Wiener algebras, algebras of holomorphic semi-almost periodic functions, and algebras of bounded holomorphic functions on Riemann surfaces.Comment: 32 pages; some explanatory comments and Example 5.1 added in Section

Boxicity and Interval-Orders: Petersen and the Complements of Line Graphs

The boxicity of a graph is the smallest dimension $d$ allowing a representation of it as the intersection graph of a set of $d$-dimensional axis-parallel boxes. We present a simple general approach to determining the boxicity of a graph based on studying its ``interval-order subgraphs''. The power of the method is first tested on the boxicity of some popular graphs that have resisted previous attempts: the boxicity of the Petersen graph is $3$, and more generally, that of the Kneser-graphs $K(n,2)$ is $n-2$ if $n\ge 5$, confirming a conjecture of Caoduro and Lichev [Discrete Mathematics, Vol. 346, 5, 2023]. Since every line graph is an induced subgraph of the complement of $K(n,2)$, the developed tools show furthermore that line graphs have only a polynomial number of edge-maximal interval-order subgraphs. This opens the way to polynomial-time algorithms for problems that are in general $\mathcal{NP}$-hard: for the existence and optimization of interval-order subgraphs of line-graphs, or of interval-completions of their complement.Comment: 17 pages, 5 figures, appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023

Graph Theory

Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures