Let K_n be the complete biorientation of K_n . Given a coloring of the arcs of K_n , a subdigraph D of K_n will be called rainbow if no two arcs of D receive the same color. For an integer n, let ar(n, C_4) be the minimum integer k such that every coloring of the arcs of K_n with k colors produces a rainbow copy of the directed cycle of order 4. In this paper, we show that for n≥ 9 , ar(n, C_4)=τ (n,4)+2 , where τ (n,4) is the Turán number of K_4 , and characterize the extremal colorings for n≥ 12 (also bounds for the small cases are presented). Using these results it is proved that for n≥ 9 , ex(n, P_4) = τ (n,4) where ex(n, P_4) is the maximum size of a simple digraph free of directed paths of order 4 (the Turán number of P_4 ).
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.
