ABSTRACT. The House of Graphs is an online database of graphs which can be accessed at https://houseofgraphs.org/. It serves as a central repository for complete lists of graphs for various graph classes. However, its main feature is a searchable database of so-called "interesting" graphs. The development of the original House of Graphs started in 2010 and it was completely rebuilt in 2021-2022. Each graph in the database is accompanied by a significant amount of meta-data such as a name, drawings, precomputed graph invariants, and comments. Given this volume of information and the importance of reliability in the scientific world, robust data management is essential to ensure accuracy and consistency across the database. In this article, we therefore focus on knowledge management in the House of Graphs and describe the inner workings of the House of Graphs and how we ensure that its data is coherent, qualitative and stable.

ABSTRACT. Graphs, maps on surfaces, and abstract polytopes are related combinatorial structures that tend to be studied by different communities using their own tools and databases. Maniplexes provide a unifying framework that captures all of them. A single database built around maniplexes would help researchers recognize shared structures and translate results across fields. Here we present a compact, interoperable format for storing maniplexes as edge-labeled graphs, designed with such a database in mind. As a first step, we connect two existing datasets of regular 4-maniplexes to the House of Graphs and to Potočnik's tetravalent graph censuses, using canonical forms of their flag graphs, 1-skeleton graphs, and 1-coskeleton graphs.

ABSTRACT. We introduce a combinatorial rewriting model for origami that provides a formal foundation for describing folding processes. While origami constructions are inherently geometric, many of their essential features—such as face adjacency, layering, and the evolution of structure through folds— can be captured at a combinatorial level.

We define an abstract origami structure consisting of a finite set of faces equipped with adjacency and layering relations, and formalize folding as a sequence of rewrite steps composed of face division and face rotation. The update of the layering relation is defined by combining combinatorial rules with geometric conditions, in particular overlap relations induced by folding. This interaction between discrete structure and geometric constraints is central to the model.

Our approach is related to earlier work on algebraic graph rewriting in origami~\ref{see-at-the-abstract}, but the model proposed here introduces a new structural formulation and operational semantics that are more directly aligned with the current design of the Eos (e-origami) system. In particular, we formulate well-formedness as a global invariant and clarify how rewrite steps preserve consistent layering while allowing complex geometric interactions.

The proposed framework provides a basis for formal reasoning about origami constructions and offers a bridge between symbolic computation and geometric modeling. It also opens the way to systematic analysis, verification, and implementation of origami processes within computational systems.

------ Reference: Tetsuo Ida, Hidekazu Takahashi: Origami fold as algebraic graph rewriting. J. Symb. Comput. 45(4): 393-413 (2010)

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ABSTRACT. Integer sequences in the OEIS span values from single-digit constants to astronomical factorials and exponentials, making prediction challenging for standard tokenised models that cannot handle out-of-vocabulary values or exploit periodic arithmetic structure. We present IntSeqBERT, a dual-stream Transformer encoder for masked integer-sequence modelling on OEIS. Each sequence element is encoded along two complementary axes: a continuous log-scale magnitude embedding and sin/cos modulo embeddings for 100 residues (moduli 2 to 101), fused via FiLM. Three prediction heads (magnitude regression, sign classification, and modulo prediction for 100 moduli) are trained jointly on 274,705 OEIS sequences. At the Large scale (91.5M parameters), IntSeqBERT achieves 95.85% magnitude accuracy and 50.38% Mean Modulo Accuracy (MMA) on the test set, outperforming a standard tokenised Transformer baseline by +8.9 pt and +4.5 pt, respectively. An ablation removing the modulo stream confirms it accounts for +15.2 pt of the MMA gain and contributes an additional +6.2 pt to magnitude accuracy. A probabilistic Chinese Remainder Theorem (CRT)-based Solver converts the model's predictions into concrete integers, yielding a 7.4-fold improvement in next-term prediction over the tokenised-Transformer baseline (Top-1: 19.09% vs. 2.59%). Modulo spectrum analysis reveals a strong negative correlation between Normalised Information Gain (NIG) and Euler's totient ratio phi(m)/m (r = -0.851, p < 10^{-28}), providing empirical evidence that composite moduli capture OEIS arithmetic structure more efficiently via CRT aggregation.