Existence of orientably-regular but chiral maps of arbitrary hyperbolic type is known as a consequence of a general theorem of Jones [12] and more specific theorems by Conder et al. [8], with proofs relying respectively on holomorphic differentials and permutation groups defined by coset diagrams. An extension to existence of orientably-regular maps of any given hyperbolic type with no exponent except 1 was obtained recently by Bachratá and Bachratý [3] with the help of canonical covers of maps. With help of parallel products of maps, we give a short proof of the latter extension. Moreover, using maps on linear fractional groups we also establish existence of non-orientable regular maps of an arbitrary hyperbolic type with no exponent except ± 1 .
The aim of this note is to prove that maximum k -forest, in the sense of Lovász, are tight, in the sense of Arocha, Bracho and Neumann-Lara. This solves a conjecture of Victor Neumann-Lara.
We give two direct bijections between the sets of 3412-avoiding and 4321-avoiding involutions of length n, respectively, and the set of (n,n+1,n+2) -core partitions. Our first bijection is very explicit and relates the 2-cycles in an involution without crossings (a 3412-avoiding involution) to the abacus of the corresponding core partition. With the second bijection, we answer a question of Amdeberhan posed in 2022. The same bijection relies on an interesting correspondence (alluded in a work of Amdeberhan and Leven) between Motzkin paths with n steps and the lower ideals of the poset comprised of the integers that are not positive linear combinations of n,n+1 and n+2 . The numbers in such a lower ideal for a Motzkin path corresponding to an involution without nestings (a 4321-avoiding involution) give the beta set of a core partition.
The Lotka–Volterra system is the simplest model of the ecological interactions of n species. The sign pattern of its parameter space ℝ^n×ℝ^n× n defines the network structure of the competitive, mutualistic, and predator–prey interactions between these species. Here, we study the feasible and stable equilibria of the Lotka–Volterra system from the perspective of computational algebraic geometry. The feasibility and stability conditions stratify ℝ^n×ℝ^n× n into feasible-stable semialgebraic sets. We encode them on the real Grassmannian Gr_ℝ(n,2n) via a parameter matrix representation, and use oriented matroid theory to develop an algorithm, combining Grassmann–Plücker relations with branching under feasibility and stability constraints. This symbolic approach determines whether a given sign pattern in ℝ^n×ℝ^n× n admits a consistent extension to Plücker coordinates. As an application, we establish the impossibility of certain interaction networks, showing that the corresponding patterns admit no such extension satisfying feasibility and stability conditions, through an effective implementation. We complement these results using numerical nonlinear algebra with HypersurfaceRegions.jl to decompose the parameter space and detect rare feasible-stable sign patterns.
We describe the Coxeter permutahedra, recently studied by Ardila, Castillo, Eur and Postnikov, in terms of random Coxeter tournaments, which involve cooperative and solitaire games, as well as the usual competitive games in graph tournaments. In this way, we establish a Coxeter version of Moon’s theorem on random tournaments. We present a geometric proof by the Mirsky–Thompson generalized Birkhoff’s theorem, a probabilistic proof by Strassen’s coupling theorem, and an algorithmic proof by a Coxeter analogue of the Havel–Hakimi algorithm. These proofs have interpretations in terms of players choosing competitors/collaborators with respect to relative weakness/strength. We also introduce a natural Coxeter analogue of the Bradley–Terry model, from the statistical theory of paired comparisons.
The rotation graph ℛ(G) is the graph whose vertices correspond to search trees on a graph G, with edges determined by rotation operations. In this paper, we analyze how the structure of a rotation graph changes when certain operations are applied to the underlying graph G. Specifically, we examine the effects of three key operations: adding a pendant vertex, adding a true twin to a vertex, and adding a false twin to a vertex. For each of these operations, we provide a full structural characterization of the new rotation graph. Using these descriptions, we investigate the chromatic number of rotation graphs, identifying conditions under which this parameter remains unchanged. As an application, we show that the chromatic number of the rotation graphs of non-complete threshold graphs (including complete split graphs and star graphs) and complete bipartite graphs is 3.
Given a dissimilarity d on a n-set X, a tree T with vertex set X is said to be R-compatible with (X, d) if for all x, z ∈ X and y on the path (in T) between x and z, we have d(x, z) ≥max{d(x, y), d(y, z) . If T is R-compatible with (X, d) , then T is a minimum spanning of (X, d) . We say that (X, d) is tree-Robinson if all its minimum spanning trees are R-compatible. In this paper, we give a local characterization of these dissimilarities, in the sense that, although the definition of tree-Robinson dissimilarities involves all minimum spanning trees of (X, d) , our characterization involves only some of them. This yields an efficient O(n^3) algorithm to recognize these dissimilarities.
A chain is defined as a directed acyclic graph (DAG) with one source and one sink, where the children are ordered and the spanning tree computed using a depth-first search is a path. Such DAGs emerge in the context of tree compression and are therefore uniquely associated with a tree. The tree size of a DAG is defined as the size of the associated tree. For fixed out-degree k ≥ 2, we compute the asymptotic expected decompressed tree size of a chain of size n chosen uniformly at random, and we show that it contains a stretched exponential term of the form e^c √(n). This result also has implications for the limit distribution of Brauer chains of fixed length.
We resolve the open problem of characterizing the Frobenius number g(A) for shifted square sequences A = (a, a+1^2, … , a+k^2) with positive integer a, confirming a conjecture of Einstein et al. (Integers 7:A15, 2007). By combining a combinatorial reduction to an optimization problem with Lagrange’s Four-square theorem and generating function techniques, we derive a semi-explicit formula for g(A): a piecewise quadratic polynomial in a, classified by residue classes modulo k^2 .
We prove divisibility results concerning the number of inequivalent irreducible complex representations of certain finite groups whose degrees are not divisible by a prime p. In particular, we study the symmetric group S_n , wreath product groups of the form G ≀ S_n for a finite group G, and index 2 subgroups of these groups. Our proofs use the combinatorics of rim hook tableaux, p-cores and p-quotients of integer partitions.
The burning and forcing processes are both instances of propagation processes on graphs that are commonly used to model real-world spreading phenomena. The contribution of this paper is twofold. We first establish a connection between these two propagation processes via hypergraphs. We do so by showing a sharp upper bound on the zero forcing number of the incidence graph of a hypergraph in terms of the lazy burning number of the hypergraph, which builds up on and improves a result by Bonato et al. (Theor Comput Sci 1056:115529, 2025). Secondly, we deepen the understanding of the role of the burning process in the context of graph spectral characterizations, whose goal is to understand which graph properties are encoded in the spectrum. While for several graph properties, including the zero forcing number, it is known that the spectrum does not encode them, this question remained open for the burning number. We solve this problem by constructing infinitely many pairs of cospectral graphs which have a different burning number.
We characterise sets of points of exceptional Lie incidence geometries, that is, the natural geometries arising from spherical buildings of exceptional types 𝖥_4 , 𝖤_6 , 𝖤_7 , 𝖤_8 and 𝖦_2 , that form a line using the opposition relation. With that, we obtain a classification of so-called “geometric lines” in many of these geometries. Furthermore, our results lead to a characterisation of geometric lines in finite exceptional Lie incidence geometries as minimal blocking sets, that is, point sets of the size of a line admitting no object opposite to all of their members, in most cases, and we classify all exceptions. As a further consequence, we obtain a characterisation of automorphisms of exceptional spherical buildings as certain opposition preserving maps.
We present a counterexample to the order relation between the largest common subtree and the smallest common supertree as claimed by Rosselló and Valiente (Theor Comput Sci 362(1–3):33–53, 2006). Furthermore, we demonstrate that knowing the solution to one of these problems does not substantially simplify the computation of the other; solving the second problem still requires processing information linear in the size of the original trees.
In this paper, we investigate three fundamental problems regarding cut complexes of graphs: realizability, uniqueness of reconstruction, and algorithmic recognition. We introduce the parameter m(d, n), defined as the minimum number of additional vertices required to realize any pure d-dimensional simplicial complex on n vertices as a cut complex, and establish foundational bounds. Furthermore, we characterize precisely when a graph on n ≥ 5 vertices is uniquely reconstructible from its 3-cut complex. Based on this characterization, we develop an O(n^4) recognition algorithm. These results strengthen the connection between graph structure and the topology of cut complexes.
The generating polynomial of all n-permutations with respect to the number of alternating runs possesses a root at -1 of multiplicity ⌊ (n-2)/2 ⌋ for n ≥ 2 . This fact can be deduced by combining the David–Barton formula for Eulerian polynomials with the Foata–Schützenberger γ -decomposition of these polynomials. Recently, Bóna provided a group—action proof of this result. In the present paper, we propose an alternative approach based on the Hetyei–Reiner action on binary trees, which yields a new combinatorial interpretation of Bóna’s quotient polynomial. Furthermore, we extend our study to analogous results for permutations of types B and D. As a consequence of our bijective framework, we also obtain combinatorial proofs of David–Barton type identities for permutations of types A and B.
Phylogenetic trees and networks are graphs used to model evolutionary relationships, with trees representing strictly branching histories and networks allowing for events in which lineages merge, called reticulation events. While the question of data sufficiency has been studied extensively in the context of trees, it remains largely unexplored for networks. In this work we take a first step in this direction by establishing bounds on the amount of genomic data required to reconstruct binary level-1 semi-directed phylogenetic networks, which are binary networks in which reticulation events are indicated by directed edges, all other edges are undirected, and cycles are vertex-disjoint. For this class, methods have been developed recently that are statistically consistent. Roughly speaking, such methods are guaranteed to reconstruct the correct network assuming infinitely long genomic sequences. Here we consider the question whether networks from this class can be uniquely and correctly reconstructed from finite sequences. Specifically, we present an inference algorithm that takes as input genetic sequence data, and demonstrate that the sequence length sufficient to reconstruct the correct network with high probability, under the CFN model of evolution, scales logarithmically, polynomially, or polylogarithmically with the number of taxa, depending on the parameter regime. As part of our contribution, we also present novel inference rules for quartet data in the semi-directed phylogenetic network setting.
We adapt the vertical and horizontal insertion encodings of Cayley permutations to enumerate restricted growth functions, which are in bijection with unordered set partitions. For both insertion encodings, we fully classify the classes for which these languages are regular. For the horizontal insertion encoding, we also prove that the conditions to be regular are the same for restricted growth functions of matchings.
An n-dimensional lattice polytope 𝒬_σ can be associated to any composition σ of a positive integer n, as a special case of constructions due to Pitman–Stanley and Chapoton. The entries of the h-vector of σ , introduced by Chapoton, enumerate the lattice points in 𝒬_σ by the number of their nonzero coordinates. Chapoton conjectured that this vector is equal to the h-vector of a flag simplicial polytope. This paper proves this conjecture. Moreover, it shows that the gamma-vector associated to the h-vector of σ is nonnegative by means of an explicit combinatorial interpretation and confirms certain other conjectures of Chapoton on the lattice point enumeration of composition polytopes. A combinatorial interpretation of their h^* -polynomials is deduced.
The d-distance p-packing domination number γ _d^p(G) of a graph G is the cardinality of a smallest set of vertices of G which is both a d-distance dominating set and a p-packing. If no such set exists, then we set γ _d^p(G) = ∞ . For an arbitrary strong product G⊠ H it is proved that γ _d^p(G⊠ H) ≤γ _d^p(G) γ _d^p(H) . By proving that γ _d^p(P_m ⊠ P_n) = ⌈m/2d+1⌉⌈n/2d+1⌉ , and that if γ _d^p(C_n) < ∞ , then γ _d^p(P_m ⊠ C_n) = ⌈m/2d+1⌉⌈n/2d+1⌉ , the sharpness of the upper bound is demonstrated. On the other hand, infinite families of strong toruses are presented for which the strict inequality holds. For instance, we present strong toruses with difference 2 and demonstrate that the difference can be arbitrarily large if only one factor is a cycle. It is also conjectured that if γ _d^p(G) = ∞ , then γ _d^p(G⊠ H) = ∞ for every graph H. Several results are proved which support the conjecture, in particular, if γ _d^p(C_m)= ∞ , then γ _d^p(C_m ⊠ C_n)=∞ .
