Jan Goedgebeur

Marien Abreu, Jan Goedgebeur, Jorik Jooken et al.

A graph is pseudo 2-factor isomorphic if all of its 2-factors have the same parity of number of cycles. Abreu et al. [J. Comb. Theory, Ser. B. 98 (2008) 432--442] conjectured that $K_{3,3}$, the Heawood graph and the Pappus graph are the only essentially 4-edge-connected pseudo 2-factor isomorphic cubic bipartite graphs. This conjecture was disproved by Goedgebeur [Discr. Appl. Math. 193 (2015) 57--60] who constructed a counterexample $\mathcal{G}$ (of girth 6) on 30 vertices. Using a computer search, he also showed that this is the only counterexample up to at least 40 vertices and that there are no counterexamples of girth greater than 6 up to at least 48 vertices. In this manuscript, we show that the Gray graph -- which has 54 vertices and girth 8 -- is also a counterexample to the pseudo 2-factor isomorphic graph conjecture. Next to the graph $\mathcal{G}$, this is the only other known counterexample. Using a computer search, we show that there are no smaller counterexamples of girth 8 and show that there are no other counterexamples up to at least 42 vertices of any girth. Moreover, we also verified that there are no further counterexamples among the known censuses of symmetrical graphs. Recall that a graph is 2-factor Hamiltonian if all of its 2-factors are Hamiltonian cycles. As a by-product of the computer searches performed for this paper, we have verified that the $2$-factor Hamiltonian conjecture of Funk et al. [J. Comb. Theory, Ser. B. 87(1) (2003) 138--144], which is still open, holds for cubic bipartite graphs of girth at least 8 up to 52 vertices, and up to 42 vertices for any girth.

Gauvain Devillez, Sven D'hondt, Jan Goedgebeur

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.

Jan Goedgebeur, Jarne Renders, Steven Van Overberghe

We present an algorithm for the efficient generation of all pairwise non-isomorphic cycle permutation graphs, i.e. cubic graphs with a $2$-factor consisting of two chordless cycles, non-hamiltonian cycle permutation graphs and permutation snarks, i.e. cycle permutation graphs that do not admit a $3$-edge-colouring. This allows us to generate all cycle permutation graphs up to order $34$ and all permutation snarks up to order $46$, improving upon previous computational results by Brinkmann et al. Moreover, we give several improved lower bounds for interesting permutation snarks, such as for a smallest permutation snark of order $6 \bmod 8$ or a smallest permutation snark of girth at least $6$ and give more evidence in support of a conjecture of Goddyn. These computational results also allow us to complete a characterisation of the orders for which non-hamiltonian cycle permutation graphs exist, answering an open question by Klee from 1972, and yield many more counterexamples to conjectures by Jackson and Zhang.

Louis Stubbe, Jens Goemaere, Jan Goedgebeur

Automated guided vehicles (AGVs) are widely used in various industries, and scheduling and routing them in a conflict-free manner is crucial to their efficient operation. We propose a loop-based algorithm that solves the online, conflict-free scheduling and routing problem for AGVs with any capacity and ordered jobs in loop-based graphs. The proposed algorithm is compared against an exact method, a greedy heuristic and a metaheuristic. We experimentally show, using theoretical and real instances on a model representing a real manufacturing plant, that this algorithm either outperforms the other algorithms or gets an equally good solution in less computing time.