A mixed graph is a graph with some directed edges and some undirected edges. We introduce the notion of mixed matroids as a generalization of mixed graphs. A mixed matroid can be viewed as an oriented matroid in which the signs over a fixed subset of the ground set have been forgotten. We extend to mixed matroids standard definitions from oriented matroids, establish basic properties, and study questions regarding the reorientations of the unsigned elements. In particular we address in the context of mixed matroids the P-connectivity and P-orientability issues which have been recently introduced for mixed graphs.
Fleischner introduced the idea of splitting a vertex of degree at least three in a connected graph and used the operation to characterize Eulerian graphs. Raghunathan et. al. extended the splitting operation from graphs to binary matroids. It has been studied that splitting operation, in general, may not preserve the connectedness of the binary matroid. Interestingly, it is true that the splitting matroid of a disconnected matroid may be connected. In this paper, we characterize the binary disconnected matroids whose splitting matroid is connected.
We consider the class of graphs for which the edge connectivity is equal to the maximum number of edge-disjoint spanning trees, and the natural generalization to matroids, where the cogirth is equal to the number of disjoint bases. We provide descriptions of such graphs and matroids, showing that such a graph (or matroid) has a unique decomposition. In the case of graphs, our results are relevant for certain communication protocols.
Robust subsets of matroids were introduced by Huang and Sellier to propose approximate kernels for the matroid-constrained maximum vertex cover problem. In this paper, we prove that the bound for robust subsets of transversal matroids given by Huang and Sellier can be improved.
For any matroid $M$, we compute the Tutte polynomial $T_M(x,y)$ using the mixed intersection numbers of certain classes in the combinatorial Chow ring $A^\bullet(M)$ arising from hypersimplices. Using the mixed Hodge-Riemann relations, we deduce a strengthening of the log-concavity of the $h$-vector of a matroid complex, improving on an old conjecture of Dawson.
The present note is a strengthening of a recent paper by K. Takazawa and Y. Yokoi (A generalized-polymatroid approach to disjoint common independent sets in two matroids, Discrete Mathematics (2019)). For given two matroids on $E$, under the same assumption in their paper to guarantee the existence of a partition of $E$ into $k$ common independent sets of the two matroids, we show that there exists a nearly uniform partition $\mathcal{P}$ of $E$ into $k$ common independent sets, where the difference of the cardinalities of any two sets in $\mathcal{P}$ is at most one.
In this note we investigate some matroid minor structure results. In particular, we present sufficient conditions, in terms of {\em triangles}, for a matroid to have either $U_{2,4}$ or $F_7$ or $M(K_5)$ as a minor.
Given a mixed hypergraph $\mathcal{F}=(V,\mathcal{A}\cup \mathcal{E})$, functions $f,g:V\rightarrow \mathbb{Z}_+$ and an integer $k$, a packing of $k$ spanning mixed hyperarborescences is called $(k,f,g)$-flexible if every $v \in V$ is the root of at least $f(v)$ and at most $g(v)$ of the mixed hyperarborescences. We give a characterization of the mixed hypergraphs admitting such packings. This generalizes results of Frank and, more recently, Gao and Yang. Our approach is based on matroid intersection, generalizing a construction of Edmonds. We also obtain an algorithm for finding a minimum we…
This note generalizes a result contained in a previous paper [ J. Sanders, Circuit preserving edge maps II, J. Combin. Theory Ser. B 42 (1987), 146-155].
A Note on an Analytic Approach to the Problem of Finite Matroid Representability, The Cardinality of Sets of k-Independent Vectors over Finite Fields and the Maximum Distance Separable Conjecture
Department of Mathematics, FIZ Karlsruhe Leibnitz Institute for Information Infrastructure（FIZ Karlsruhe莱布尼茨信息基础设施研究所数学系）；Department of Economics, University of Toronto（多伦多大学经济系）
In this short note we make a few remarks on a class of generalized incidence matrices whose matroids do not depend on the orientation of the underlying graph and natural commutative algebras associated to such matrices.
In this note we introduce a sufficient condition for the Orlik-Solomon algebra associated to a matroid M to be l-adic and we prove that this condition is necessary when M is binary (in particular graphic). Moreover, this result cannot be extended to the class of all matroids.
New families of matroids are constructed in this note. These new fami- lies are derived from the concept of linearly independent set family (LISF) introduced by Eicker and Ewald [Linear Algebra and its Applications 388 (2004) 173-191]. The proposed construction generalizes in a natural way the well known class of vectorial matroids over a field.
This note contributes to the structure theory of abstract rigidity matroids in general dimension. In the spirit of classical matroid theory, we prove several cryptomorphic characterizations of abstract rigidity matroids (in terms of circuits, cocircuits, bases, hyperplanes). Moreover, the study of hyperplanes in abstract rigidity matroids leads us to state (and support with significant evidence) a conjecture about characterizing the class of abstract rigidity matroids by means of certain "prescribed substructures". We then prove a recursive version of this conjecture.
The Lagrangian geometry of matroids was introduced in [ADH20] through the construction of the conormal fan of a matroid M. We used the conormal fan to give a Lagrangian-geometric interpretation of the h-vector of the broken circuit complex of M: its entries are the degrees of the mixed intersections of certain convex piecewise linear functions $γ$ and $δ$ on the conormal fan of M. By showing that the conormal fan satisfies the Hodge-Riemann relations, we proved Brylawski's conjecture that this h-vector is a log-concave sequence. This sequel explores the Lagrangian combinatorics of matroids, fu…
We observe that for planar graphs, the geometric duality relation generates both 2-isomorphism and abstract duality. This observation has the surprising consequence that for links, the equivalence relation defined by isomorphisms of checkerboard graphs is the same as the equivalence relation defined by 2-isomorphisms of checkerboard graphs.
We show that the algebraic rank of divisors on certain graphs is related to the realizability problem of matroids. As a consequence, we produce a series of examples in which the algebraic rank depends on the ground field. We use the theory of metrized complexes to show that equality between the algebraic and combinatorial rank is not a sufficient condition for smoothability of divisors, thus giving a negative answer to a question posed by Caporaso, Melo, and the author.
Király in [On maximal independent arborescence packing, SIAM J. Discrete. Math. 30 (4) (2016), 2107-2114] solved the following packing problem: Given a digraph $D = (V, A)$, a matroid $M$ on a set $S = \{s_{1}, \ldots,s_{k} \}$ along with a map $π: S \rightarrow V$, find $k$ arc-disjoint maximal arborescences $T_{1}, \ldots ,T_{k}$ with roots $π(s_{1}), \ldots ,π(s_{k})$, such that, for any $v \in V$, the set $\{s_{i} : v \in V(T_{i})\}$ is independent and its rank reaches the theoretical maximum. In this paper, we give a new characterization for packing of maximal independent mixed arborescen…
Tropical oriented matroids were defined by Ardila and Develin in 2007. They are a tropical analogue of classical oriented matroids in the sense that they encode the properties of the types of points in an arrangement of tropical hyperplanes -- in much the same way as the covectors of (classical) oriented matroids describe the types in arrangements of linear hyperplanes. Ardila and Develin proved that tropical oriented matroids can be represented as mixed subdivisions of dilated simplices. In this paper we show that this correspondence is a bijection. Moreover, a tropical analogue for the Topol…
In this note we generalize the convolution formula for the Tutte polynomial of Kook-Reiner-Stanton and Etienne-Las Vergnas to a more general setting that includes both arithmetic matroids and delta-matroids. As corollaries, we obtain new proofs of two positivity results for pseudo-arithmetic matroids and a combinatorial interpretation of the arithmetic Tutte polynomial at infinitely many points in terms of arithmetic flows and colorings. We also exhibit connections with a decomposition of Dahmen-Micchelli spaces and lattice point counting in zonotopes.
The first non-trivial case of Hadwiger's conjecture for oriented matroids reads as follows. If $\mathcal{O}$ is an $M(K_4)$-free oriented matroid, then $\mathcal{O}$ admits a NZ $3$-coflow, i.e., it is $3$-colourable in the sense of Hochstättler-Nešetřil. The class of gammoids is a class of $M(K_4)$-free orientable matroids and it is the minimal minor-closed class that contains all transversal matroids. Towards proving the previous statement for the class of gammoids, Goddyn, Hochstättler, and Neudauer conjectured that every gammoid has a positive coline (equivalently, a positive double circui…
Let A be a (central) arrangement of hyperplanes in a finite dimension complex vector space V. Let M(A) be the dependence matroid determined by A. The Orlik-Solomon algebra OS(M) of a matroid M is the exterior algebra on the points modulo the ideal generated by circuit boundaries. The algebra OS(M) is isomorphic to the cohomology algebra of the complement in V of the union of the hyperplanes of A. The Tutte polynomial T(x,y) of M is a powerful invariant of the matroid M. When M(A) is a rank three matroid and A is the complexification of a real arrangement, we prove that OS(M) determines T(x,y).…
In 1908 Voronoi conjectured that every convex polytope which tiles space face-to-face by translations is affinely equivalent to the Dirichlet-Voronoi polytope of some lattice. In 1999 Erdahl proved this conjecture for the special case of zonotopes. A zonotope is a projection of a regular cube under some affine transformation. In 1975 McMullen showed several equivalent conditions for a zonotope to be a space tiling zonotope, i.e. a zonotope which admits a face-to-face tiling of space by translations. Implicitly, he related space tiling zonotopes to a special class of oriented matroids (regular …
In this note we prove a lower bound for the rank of 2-dimensional generic rigidity matroid for regular graphs of degree four and five. Also, we give examples to show the order of the bound we give is sharp.
Short rewriting, and geometric explanations related to the active bijection, for: Extension-lifting bijections for oriented matroids, by S. Backman, F. Santos, C.H. Yuen, arXiv:1904.03562v2 (October 29, 2023)
Ardila and Develin's paper on tropical oriented hyperplane arrangements and tropical oriented matroids defines tropical oriented matroids and conjectures a bijection between them and triangulations of products of simplices $Δ_{n-1} \times Δ_{d-1}$. Oh and Yoo recently confirmed this conjecture; however, neither group addressed the case of hyperplanes that are not in generic position. These non-generic arrangements do not correspond to tropical oriented matroids, but they encode information about subdivisions of $Δ_{n-1} \times Δ_{d-1}$. This note considers the non-generic case and presents som…
In this note we provide a higher-dimensional analogue of Tutte's celebrated theorem on colorings and flows of graphs, by showing that the theory of arithmetic Tutte polynomials and quasi-polynomials encompasses invariants defined for CW complexes by Beck-Breuer-Godkin-Martin and Duval-Klivans-Martin. Furthermore, we answer a question by Bajo-Burdick-Chmutov, concerning the modified Tutte-Krushkal-Renhardy polynomials defined by these authors: to this end, we prove that the product of two arithmetic multiplicity functions on a matroid is again an arithmetic multiplicity function.
We present a new proof of an algebraic characterization of circle graphs due to W. Naji. For bipartite graphs, Naji's theorem is equivalent to an algebraic characterization of planar matroids due to J. Geelen and B. Gerards. Naji's theorem also yields an algebraic characterization of permutation graphs.
For two graphs, $G$, and $H$, an edge-coloring of a complete graph is $(G,H)$-good if there is no monochromatic subgraph isomorphic to $G$ and no rainbow subgraph isomorphic to $H$ in this coloring. The set of number of colors used by some $(G,H)$-colorings of $K_n$ is called a mixed-Ramsey spectrum. This note addresses a fundamental question of whether the spectrum is an interval. It is shown that the answer is "yes" if $G$ is not a star and $H$ does not contain a pendent edge.
A note on integer programming methods for mixed radial Moore graphs
Eindhoven University of Technology（埃因霍温理工大学）；Vrije Universiteit Brussel（布鲁塞尔自由大学）；Universitat Rovira i Virgili（罗维拉-依吉利维尔大学）；Universitat de Lleida（莱里达大学）
In recent work of Cachazo, Guevara, Mizera and the author, a generalization of the biadjoint scattering amplitude $m^{(k)}(\mathbb{I}_n,\mathbb{I}_n)$ was introduced as an integral over the moduli space of $n$ points in $\mathbb{CP}^{k-1}$, with value a sum of certain rational functions on the kinematic space $\mathcal{K}_{k,n}$. It was shown there for $m^{(3)}(\mathbb{I}_6,\mathbb{I}_6)$ and later by Cachazo and Rojas that collections of poles appearing in $m^{(3)}(\mathbb{I}_7,\mathbb{I}_7)$ are compatible exactly when they are dual to collections of rays which generate the maximal faces of …
A mixed graph is cospectral to its converse, with respect to the usual adjacency matrices. Hence, it is easy to see that a mixed graph whose eigenvalues occur uniquely, up to isomorphism, must be isomorphic to its converse. It is therefore natural to ask whether or not this is a common phenomenon. This note contains the theoretical evidence to confirm that the fraction of self-converse mixed graphs tends to zero.
In this note we investigate mixed partitions with extra condition on the sizes of the blocks. We give a general formula and the generating function. We consider in more details a special case, determining the generating functions, some recurrences and a connection to r-Stirling numbers. To obtain our results, we use pure combinatorial arguments, classical manipulations of generating functions and to derive the generating functions we apply the symbolic method.
Jump systems are sets of integer vectors satisfying a simple axiom, generalizing matroids, also delta-matroids, and well-kown combinatorial examples such as degree sequences of subgraphs of a graph. It is useful to know if a set of vectors defined from combinatorial structures is a jump system: this has consequences for optimizing on the set, or on some derived sets of vectors. In this note we are mainly concerned in telling our proof of the following more than two decades old fact and its original, elementary proof for an example different from degree sequences: {\em Given an udirected graph …
This note provides a simple proof for the equality between the normalized volume of a convex polytope with $m$ vertices and the mixed volume of $m$ simplices and thus shows the seemingly restrictive problem of computing mixed volume of simplices is still at least as hard as computing volumes of convex polytopes.
