Monotone Paths Research Articles

Tiling Edge-Ordered Graphs with Monotone Paths and Other Structures

Tiling Edge-Ordered Graphs with Monotone Paths and Other Structures

Ramsey Problems for Monotone Paths in Graphs and Hypergraphs

The study of ordered Ramsey numbers of monotone paths for graphs and hypergraphs has a long history, going back to the celebrated work by Erdős and Szekeres in the early days of Ramsey theory. In this paper we obtain several results in this area, establishing two conjectures of Mubayi and Suk and improving bounds due to Balko, Cibulka, Král and Kynčl. For example, in the graph case, we show that the ordered Ramsey number for a fixed clique versus a fixed power of a monotone path of length n is always linear in n. Also, in the 3-graph case, we show that the ordered Ramsey number for a fixed clique versus a tight monotone path of length n is always polynomial in n. As a by-product, we also obtain a color-monotone version of the well-known Canonical Ramsey Theorem of Erdős and Rado, which could be of independent interest.

Ramsey numbers of cliques versus monotone paths

One formulation of the Erdős-Szekeres monotone subsequence theorem states that for any red/blue coloring of the edge set of the complete graph on {1,2,…,N}, there exists a monochromatic red s-clique or a monochromatic blue increasing path Pn with n vertices, provided N>(s−1)(n−1). Here, we prove a similar statement as above in the off-diagonal case for triple systems, with the quasipolynomial bound N>2c(logn)s−1. For the tth power Pnt of the ordered increasing graph path with n vertices, we prove a near linear bound cn(logn)s−2 which improves the previous bound that applied to a more general class of graphs than Pnt due to Conlon-Fox-Lee-Sudakov.

Monotone Paths on Cross-Polytopes

Monotone Paths on Cross-Polytopes

The Polyhedral Geometry of Pivot Rules and Monotone Paths

The Polyhedral Geometry of Pivot Rules and Monotone Paths

Green groupoids of 2-Calabi–Yau categories, derived Picard actions, and hyperplane arrangements

We present a construction of (faithful) group actions via derived equivalences in the general categorical setting of algebraic 2-Calabi–Yau triangulated categories. To each algebraic 2-Calabi–Yau category C \mathscr {C} satisfying standard mild assumptions, we associate a groupoid G C \mathscr {G}_{ \mathscr {C} } , named the green groupoid of C \mathscr {C} , defined in an intrinsic homological way. Its objects are given by a set of representatives m r i g C mrig\mathscr {C} of the equivalence classes of basic maximal rigid objects of C \mathscr {C} , arrows are given by mutation, and relations are given by equating monotone (green) paths in the silting order. In this generality we construct a homomorphsim from the green groupoid G C \mathscr {G}_{ \mathscr {C} } to the derived Picard groupoid of the collection of endomorphism rings of representatives of m r i g C mrig\mathscr {C} in a Frobenius model of C \mathscr {C} ; the latter canonically acts by triangle equivalences between the derived categories of the rings. We prove that the constructed representation of the green groupoid G C \mathscr {G}_{ \mathscr {C} } is faithful if the index chamber decompositions of the split Grothendieck groups of basic maximal rigid objects of C \mathscr {C} come from hyperplane arrangements. If Σ 2 ≅ i d \Sigma ^2 \cong id and C \mathscr {C} has finitely many equivalence classes of basic maximal rigid objects, we prove that G C \mathscr {G}_{ \mathscr {C} } is isomorphic to a Deligne groupoid of a hyperplane arrangement and that the representation of this groupoid is faithful.

Scrambled Vandermonde convolutions of Gaussian polynomials

It is well known that Gaussian polynomials (i.e., q-binomials) describe the distribution of the ▪ statistic on monotone paths in a rectangular grid. We introduce two new statistics, ▪ and ▪; attach “ornaments” to the grid that scramble the values of ▪ in specific fashion; and re-evaluate these statistics, in order to argue that all scrambled versions of the ▪ statistic are equidistributed with ▪. Our main result is a representation of the generating function for the bi-statistic ▪ as a new, two-variable Vandermonde convolution of the original Gaussian polynomial. The proof relies on explicit bijections between differently ornated paths.

On the Holt–Klee property for oriented matroid programming

The Holt–Klee theorem says that the graph of a d-polytope, with edges oriented by a linear function on P that is not constant on any edge, admits d independent monotone paths from the source to the sink. We prove that the digraphs obtained from oriented matroid programs of rank d+1 on n+2 elements, which include those from d-polytopes with n facets, admit d independent monotone paths from source to sink if d≤4. This was previously only known to hold for d≤3 and n≤6.

Non-crossing monotone paths and binary trees in edge-ordered complete geometric graphs

An edge-ordered graph is a graph with a total ordering of its edges. A path \(P=v_1v_2\ldots v_k\) in an edge-ordered graph is called increasing if \((v_iv_{i+1}) < (v_{i+1}v_{i+2})\) for all \(i = 1,\ldots,k-2\); and it is called decreasing if \((v_iv_{i+1}) > (v_{i+1}v_{i+2})\) for all \(i = 1,\ldots,k-2\). We say that P is monotone if it is increasing or decreasing. A rooted tree T in an edge-ordered graph is called monotone if either every path from the root to a leaf is increasing or every path from the root to a leaf is decreasing.Let G be a graph. In a straight-line drawing D of G, its vertices are drawn as different points in the plane and its edges are straight line segments. Let \(\overline{\alpha}(G)\) be the largest integer such that every edge-ordered straight-line drawing of G contains a monotone non-crossing path of length \(\overline{\alpha}(G)\). Let \(\overline{\tau}(G)\) be the largest integer such that every edge-ordered straight-line drawing of G contains a monotone non-crossing complete binary tree of \(\overline{\tau}(G)\) edges. In this paper we show that \(\overline \alpha(K_n) = \Omega(\log\log n)\), \(\overline \alpha(K_n) = O(\log n), \overline \tau(K_n) = \Omega(\log\log \log n)\) and \(\overline \tau(K_n) = O(\sqrt{n \log n})\).

Enumerative problems for arborescences and monotone paths on polytope graphs

AbstractEvery generic linear functional on a convex polytope induces an orientation on the graph of . From the resulting directed graph one can define a notion of ‐arborescence and ‐monotone path on , as well as a natural graph structure on the vertex set of ‐monotone paths. These concepts are important in geometric combinatorics and optimization. This paper bounds the number of ‐arborescences, the number of ‐monotone paths, and the diameter of the graph of ‐monotone paths for polytopes in terms of their dimension and number of vertices or facets.

Erdős-Hajnal-type results for monotone paths

An ordered graph is a graph with a linear ordering on its vertex set. We prove that for every positive integer k, there exists a constant ck>0 such that any ordered graph G on n vertices with the property that neither G nor its complement contains an induced monotone path of size k, has either a clique or an independent set of size at least nck. This strengthens a result of Bousquet, Lagoutte, and Thomassé, who proved the analogous result for unordered graphs.A key idea of the above paper was to show that any unordered graph on n vertices that does not contain an induced path of size k, and whose maximum degree is at most c(k)n for some small c(k)>0, contains two disjoint linear size subsets with no edge between them. This approach fails for ordered graphs, because the analogous statement is false for k≥3, by a construction of Fox. We provide some further examples showing that this statement also fails for ordered graphs avoiding other ordered trees.

On the Length of Monotone Paths in Polyhedra

Motivated by the problem of bounding the number of iterations of the simplex algorithm, we investigate the possible lengths of monotone paths followed inside the oriented graphs of polyhedra (orien...

On-line size Ramsey number for monotone [formula omitted]-uniform ordered paths with uniform looseness

An ordered hypergraph is a hypergraph H with a specified linear ordering of the vertices, and the appearance of an ordered hypergraph G in H must respect the specified order on V(G). In on-line Ramsey theory, Builder iteratively presents edges that Painter must immediately color. The t-color on-line size Ramsey numberR̃t(G) of an ordered hypergraph G is the minimum number of edges Builder needs to play (on a large ordered set of vertices) to force Painter using t colors to produce a monochromatic copy of G. The monotone tight pathPr(k) is the ordered hypergraph with r vertices whose edges are all sets of k consecutive vertices.We obtain good bounds on R̃t(Pr(k)). Letting m=r−k+1 (the number of edges in Pr(k)), we prove mt−1∕(3t)≤R̃t(Pr(2))≤tmt+1. For general k, a trivial upper bound is Rk, where R is the least number of vertices in a k-uniform (ordered) hypergraph whose t-colorings all contain Pr(k) (and is a tower of height k−2). We prove R∕(klgR)≤R̃t(Pr(k))≤R(lgR)2+ε, where ε is a positive constant and tm is sufficiently large in terms of ε−1. Our upper bounds improve prior results when t grows faster than m∕logm. We also generalize our results to ℓ-loose monotone paths, where each successive edge begins ℓ vertices after the previous edge.

$S$-Hypersimplices, Pulling Triangulations, and Monotone paths

An $S$-hypersimplex for $S \subseteq \{0,1, \dots,d\}$ is the convex hull of all $0/1$-vectors of length $d$ with coordinate sum in $S$. These polytopes generalize the classical hypersimplices as well as cubes, crosspolytopes, and halfcubes. In this paper we study faces and dissections of $S$-hypersimplices. Moreover, we show that monotone path polytopes of $S$-hypersimplices yield all types of multipermutahedra. In analogy to cubes, we also show that the number of simplices in a pulling triangulation of a halfcube is independent of the pulling order.

Nearly-linear monotone paths in edge-ordered graphs

How long a monotone path can one always find in any edge-ordering of the complete graph Kn? This appealing question was first asked by Chvátal and Komlós in 1971, and has since attracted the attention of many researchers, inspiring a variety of related problems. The prevailing conjecture is that one can always find a monotone path of linear length, but until now the best known lower bound was n2/3-o(1). In this paper we almost close this gap, proving that any edge-ordering of the complete graph contains a monotone path of length n1-o(1).

Degradation modeling of 2 fatigue‐crack growth characteristics based on inverse Gaussian processes: A case study

AbstractMost modern products that are highly reliable are complex in their inner and outer structures. This situation indicates quality characterization by the interaction of multiple performance characteristics, which motivates the utilization of robust reliability models to obtain robust estimates. It is paramount to obtaining substantial information about a product's life cycle; therefore, when multiple performance characteristics are dependent, it is important to find models that address the joint distribution of performance degradation of such. In this paper, a reliability model for products with 2 fatigue‐crack growth characteristics related to 2 degradation processes is developed. The proposed model considers the dependence among degradation processes by using copula functions considering the marginal degradation processes as inverse Gaussian processes. The statistical inference is performed by using a Bayesian approach to estimate the parameters of the joint bivariate model. A time‐scale transformation is considered to assure monotone paths of the degradation trajectories. The comparison results of the reliability analysis, under both dependent and independent assumptions, are reported with the implementation of the proposed modeling in a case study, which consists of the crack propagation data of 2 terminals of an electronic device.

Monotone Paths in Geometric Triangulations

(I) We prove that the (maximum) number of monotone paths in a triangulation of n points in the plane is \(O(1.8027^n)\). This improves an earlier upper bound of \(O(1.8393^n)\); the current best lower bound is \(\varOmega (1.7034^n)\). (II) Given a planar straight-line graph G with n vertices, we show that the number of monotone paths in G can be computed in \(O(n^2)\) time.

Spanning cactus: Complexity and extensions

Minimum spanning cactus and minimum spanning cactus extension problems are studied. Both problems are NP-Complete. We present an approximation algorithm for the minimum spanning cactus extension of a forest, a hardness of approximation result of the general minimum spanning cactus problem. For the minimum spanning cactus extension problem, Kabadi and Punnen presented polynomial time algorithms for extending quasi-stars, spanning trees (Kabadi and Punnen, 2013). We present improved analysis of their algorithms in both cases. We further show that their algorithm for the extension of spanning trees can be generalized to extend any connected spanning partial cactus. As a requirement of improved implementation, we have presented a new O(n3) algorithm to compute all minimum cost monotone paths with respect to a spanning tree.

Signature inversion for monotone paths

The aim of this article is to provide a simple sampling procedure to reconstruct any monotone path from its signature. For every $N$, we sample a lattice path of $N$ steps with weights given by the coefficient of the corresponding word in the signature. We show that these weights on lattice paths satisfy the large deviations principle. In particular, this implies that the probability of picking up a “wrong” path is exponentially small in $N$. The argument relies on a probabilistic interpretation of the signature for monotone paths.

Ramsey numbers for degree monotone paths

A path v1,v2,…,vm in a graph G is degree-monotone if deg(v1)≤deg(v2)≤⋯≤deg(vm) where deg(vi) is the degree of vi in G. Longest degree-monotone paths have been studied in several recent papers. Here we consider the Ramsey type problem for degree monotone paths. Denote by Mk(m) the minimum number M such that for all n≥M, in any k-edge coloring of Kn there is some 1≤j≤k such that the graph formed by the edges colored j has a degree-monotone path of order m. We prove several nontrivial upper and lower bounds for Mk(m).