Related papers: Pseudo-orientable ribbon graphs: Matrix--Quasi-tre…
Matroids over tracts provide an algebraic framework simultaneously generalizing the notions of matroids, oriented matroids, and valuated matroids, presented by Baker and Bowler. Pendavingh partially extended this theory to skew hyperfields…
Consider a collection of points in the plane and the sets of slopes or directions of the lines between pairs of points. It is known that the algebraic matroid on the set of direction constraints between the points is equivalent to the…
In his pioneering paper on matroids in 1935, Whitney obtained a characterization for binary matroids and left a comment at end of the paper that the problem of characterizing graphic matroids is the same as that of characterizing matroids…
Delta-matroid theory is often thought of as a generalization of topological graph theory. It is well-known that an orientable embedded graph is bipartite if and only if its Petrie dual is orientable. In this paper, we first introduce the…
Using a known recursive formula for the Grothendieck classes of the moduli spaces $\overline{\mathcal M}_{0,n}$, we prove that they satisfy an asymptotic form of ultra-log-concavity as polynomials in the Lefschetz class. We also observe…
Motivated by the problem of Hurwitz equivalence of $\Delta ^2$ factorization in the braid group, we address the problem of Hurwitz equivalence in the symmetric group, obtained by projecting the $\Delta ^2$ factorizations into $S_n$. We get…
We characterize the shifted simple graphs and the $3$-uniform shifted hypergraphs whose inverse image under exterior shifting is the set of bases of a matroid: those are exactly the hypergraphs whose hyperedges form an initial lex-segment.…
We give two graph theoretical characterizations of tope graphs of (complexes of) oriented matroids. The first is in terms of excluded partial cube minors, the second is that all antipodal subgraphs are gated. A direct consequence is a third…
Vassiliev (finite type) invariants of knots can be described in terms of weight systems. These are functions on chord diagrams satisfying so-called 4-term relations. In the study of the sl2 weight system, it was shown that its value on a…
A half-tree is an edge configuration whose superimposition with a perfect matching is a tree. In this paper, we prove a half-tree theorem for the Pfaffian principal minors of a skew-symmetric matrix whose column sum is zero; introducing an…
Chordal graphs are important in the structural and algorithmic graph theory. A digraph analogue of chordal graphs was introduced by Haskin and Rose in 1973 but has not been a subject of active studies until recently when a characterization…
We characterize which systems of sign vectors are the cocircuits of an oriented matroid in terms of the cocircuit graph.
We associate root polytopes to directed graphs and study them by using ribbon structures. Most attention is paid to what we call the semi-balanced case, i.e., when each cycle has the same number of edges pointing in the two directions.…
Cliquewidth is a dense analogue of treewidth. It can be deduced from recent results by Hickingbotham [arXiv:2501.10840] and Nguyen, Scott, and Seymour [arXiv:2501.09839] that graphs of bounded cliquewidth are quasi-isometric to graphs of…
We consider the problem of classifying those graphs that arise as an undirected square of an oriented graph by generalising the notion of quasi-transitive directed graphs to mixed graphs. We fully classify those graphs of maximum degree…
A biased graph is a graph with a class of selected circles ("cycles", "circuits"), called "balanced", such that no theta subgraph contains exactly two balanced circles. A biased graph has two natural matroids, the frame matroid and the lift…
The Topological Representation Theorem for (oriented) matroids states that every (oriented) matroid can be realized as the intersection lattice of an arrangement of codimension one homotopy spheres on a homotopy sphere. In this paper, we…
We establish a connection between the algebraic geometry of the type B permutohedral toric variety and the combinatorics of delta-matroids. Using this connection, we compute the volume and lattice point counts of type B generalized…
We prove that every oriented tree on $n$ vertices with bounded maximum degree appears as a spanning subdigraph of every directed graph on $n$ vertices with minimum semidegree at least $n/2+o(n)$. This can be seen as a directed graph…
We introduce a new group action on set systems, constructed as a semidirect product of a permutation group and a group generated by twist and loop complementation operations on a single element. This action extends the ribbon group…