arXiv subjects

Meike Weiß

Publications and source records attributed to Meike Weiß.

10 recordsLinked to original sources

arXiv2026

Strong Embeddings of 3-Connected Cubic Planar Graphs on Surfaces of non-negative Euler Characteristic

Whitney proved that 3-connected planar graphs admit a unique embedding on the sphere. In contrast, Enami investigated embeddings of 3-connected cubic planar graphs on non-spherical surfaces with non-negative Euler characteristic. He established that such an embedding exists if and only if the dual graph contains a particular subgraph. Here, strong embeddings are investigated motivated by the cycle double cover conjecture and the relation to triangulated surfaces. We provide a complete characterization of strong embeddings on the projective plane, the torus, and the Klein bottle in terms of a distinguished subset of Enami's subgraphs. This characterization not only deepens the structural understanding of graph embeddings on non-spherical surfaces, but also establishes a robust foundation for computing cycle double covers. As a direct consequence, we derive explicit criteria that determine when a graph does not admit a strong embedding on these surfaces-offering new tools for both theoretical analysis and algorithmic applications.

math.CO↗

arXiv2026

Polyhedral Maps of Cubic Graphs with given Automorphism Groups

L. Babai introduced a method for constructing a cubic graph whose automorphism group is isomorphic to a given finite group $G$, obtained by modifying a corresponding Cayley graph of $G$. Building on this approach, we construct a cubic graph that admits a polyhedral map whose automorphism group, as well as the automorphism group of the polyhedral map itself, is isomorphic to $G$.

math.CO↗

arXiv2026

Strong Embeddings of Regular Graphs with Prescribed Automorphism Groups

A classical theorem of Frucht states that every finite group occurs as the automorphism group of a finite graph. We prove an embedded analogue for regular graphs of arbitrary degree. In particular, we show that for every $d\geq 3$ and every finite group $G$, there exists a $d$-regular graph $Γ$ with a strong embedding $β$ such that $\mathrm{Aut}(Γ) \cong \mathrm{Aut}(β(Γ)) \cong G.$ Further, we prove that for every such $d$ and $G$ there exists a sequence of $d$-regular graphs with corresponding strong embeddings whose genera form an unbounded sequence and whose automorphism groups are isomorphic to $G$. Along the way, we identify an oversight in Sabidussi's classical construction of regular graphs with prescribed automorphism group. We give an alternative construction that corrects this issue and strengthens Sabidussi's result by producing an automorphism group-invariant proper $d$-edge-colouring.

math.CO↗

arXiv2026

Construction Methods for Space-Filling Heterogeneous Topological Interlocking Assemblies

Deforming fundamental domains of wallpaper groups provides a systematic way to generate non-convex blocks which admit topological interlocking assemblies (TIAs). We use this approach to construct TIAs that fully occupy the space between two parallel planes and incorporate multiple block types. In addition to wallpaper groups, semiregular tessellations are employed in the construction of such TIAs. These construction methods open up an extensive design space for TIAs, expanding the possibilities of feasible interlocking systems and creating new opportunities for architectural and material design. Several resulting block families can be interpreted as geometric realizations of generalized Truchet tiles or decorated lozenge tilings and, with suitable colouring rules, we establish a one-to-one correspondence between these tilings and specific TIAs. This framework enables a systematic investigation of symmetric and asymmetric assemblies derived from diverse block types.

math.GR↗

arXiv2026

Influence of the geometry on the mechanical performance of tubular interlockings: A study of the Sine Block

Topological interlocking assemblies (TIA) are arrangements of blocks such that rigid-body motions of the blocks are fully constrained by their neighbours and a fixed frame. In this work, we investigate tubular interlocking structures derived from the sine curve and parametrised by several geometric design parameters. We analyse the behaviour of these parametrised tubular interlockings under various boundary conditions and examine how our proposed parameters influence the mechanical response. For this purpose, we first develop a simplified multibody dynamics formulation that enables an efficient exploration of how the design parameters of the block influence the load transfer within the assembly. To further corroborate these results, we perform several finite element simulations, which give insights into the mechanical behaviour of our proposed TIA. Our results show that the block geometry plays a decisive role in the mechanical performance of the corresponding TIA. We additionally discuss the problem of exploding TIAs and demonstrate that the TIA resulting from our Sine Block does not exhibit this behaviour. Lastly, we provide evidence that non-exploding TIAs possess better mechanical properties than exploding ones.

cs.CE↗

arXiv2026

On 3-Connected Planar Graphs with Unique Orientable Circuit Double Covers

A circuit double cover of a bridgeless graph is a collection of even subgraphs such that every edge is contained in exactly two subgraphs of the given collection. Such a circuit double cover describes an embedding of the corresponding graph onto a surface. In this paper, we investigate the well-known Orientable Strong Embedding Conjecture. This conjecture proposes that every bridgeless graph has a circuit double cover describing an embedding on an orientable surface. In a recent paper, we have proved that a 3-connected cubic planar graph G has exactly one orientable circuit double cover if and only if G is the dual graph of an Apollonian network. In this paper, we extend this result by demonstrating that this characterisation applies to any 3-connected planar graph, regardless of whether it is cubic.

math.CO↗

arXiv2025

Influence of a generative parameter on the mechanical performance of topological interlocking assemblies of a hexagonal block

A topological interlocking assembly is an arrangement of blocks, where all blocks are kinematically constrained by their neighboring blocks and a fixed frame. This concept has been known for a long time, attracting recent interest due to its advantageous mechanical properties, such as reusability, redundancy and limited crack propagation. New mathematical methods enable the generation of vast numbers of new topologically interlocking blocks. A natural next question is the quantification of the mechanical performance of these new blocks. We conduct a numerical study of topological interlocking assemblies whose blocks are constructed based on the hexagonal grid. By varying a design parameter used in the generation of these blocks, we study its influence on the structural performance of the entire assembly. The results improve our understanding of the link between the block parameters and the mechanical performance. This enhances the ability to custom design blocks for certain mechanical requirements of the topological interlocking assemblies.

cond-mat.mtrl-sci↗

arXiv2025

On 3-Connected Cubic Planar Graphs and their Strong Embeddings on Orientable Surfaces

Although the strong embedding of a 3-connected planar graph $G$ on the sphere is unique, $G$ can have different inequivalent strong embeddings on a surface of positive genus. If $G$ is cubic, then the strong embeddings of $G$ on the projective plane, the torus and the Klein bottle each are in one-to-one correspondence with certain subgraphs of the dual graph $G^\ast$. Here, we exploit this characterisation and show that two strong embeddings of $G$ on the projective plane, the torus or the Klein bottle are isomorphic if and only if the corresponding subgraphs of $G^{\ast}$ are contained in the same orbit under $\mathrm{Aut}(G^{\ast})$. This allows us to construct a data base containing all isomorphism classes of strong embeddings on the projective plane, the torus and the Klein bottle of all 3-connected cubic planar graphs with up to 22 vertices. Moreover, we establish that cyclically 4-edge connected cubic planar graphs can be strongly embedded on orientable surfaces of positive genera. We use this to show that a 3-connected cubic planar graph has no strong embedding on orientable surfaces of positive genera if and only if it is the dual of an Apollonian network.

math.CO↗

arXiv2025

Interplay of Cubic Graphs and Simplicial Surfaces

Simplicial surfaces describe the incidence relations between vertices, edges and faces of triangulated 2-dimensional manifolds in a purely combinatorial way. By considering only the incidences of edges and faces, simplicial surfaces are closely related to cubic graphs. In this paper we investigate how properties of simplicial surfaces and cubic graphs can be transferred to each other. Furthermore, we study embeddings of cubic graphs on simplicial surfaces and how they are connected to strong graph embeddings. For instance, 3-connected cubic planar graphs are uniquely embeddable on simplicial spheres, which is a direct consequence of Whitney's embedding theorem. Moreover, 3-connected cubic planar graphs can also be embedded on simplicial surfaces of higher genus. We characterise the properties that a simplicial sphere must possess such that the cubic graph describing its edge-face incidence relation can be embedded on a simplicial surface of non-negative Euler characteristic.

math.CO↗

arXiv2023

Mechanical Comparison of Arrangement Strategies for Topological Interlocking Assemblies

Topological Interlocking assemblies are arrangements of blocks kinematically constrained by a fixed frame, such that all rigid body motions of each block are constrained only by its permanent contact with other blocks and the frame. In the literature several blocks are introduced that can be arranged into different interlocking assemblies. In this study we investigate the influence of arrangement on the overall structural behaviour of the resulting interlocking assemblies. This is performed using the Versatile Block, as it can be arranged in three different doubly periodic ways given by wallpaper symmetries. Our focus lies on the load transfer mechanisms from the assembly onto the frame. For fast a priori evaluation of the assemblies we introduce a combinatorial model called Interlocking Flows. To investigate our assemblies from a mechanical point of view we conduct several finite element studies. These reveal a strong influence of arrangement on the structural behaviour, for instance, an impact on both the point and amount of maximum deflection. The results of the finite element analysis are in very good agreement with the predictions of the Interlocking Flow model. Our source code, data and examples are available under https://doi.org/10.5281/zenodo.10246034.

cs.CE↗