Tight Path Research Articles

New Adaptive Mesh Refinement Strategy for Entry Guidance via Sequential Convex Programming

This study focuses on the entry guidance problem, which is a highly nonlinear and constrained optimal control problem. The entry guidance problem is formulated as a nonlinear trajectory optimization problem with six-degrees-of-freedom dynamics, path constraints, and boundary constraints. The formulated problem is discretized and solved using sequential convex programming (SCP) with successive linearization. However, during the discretization process, conventional collocation methods encounter a challenge with high-frequency jittering in the control and bank angle profiles near active path constraints. Because the considered problem especially involves tight path constraints, the solution jittering issue becomes severe. Therefore, this study proposes a novel mesh refinement strategy to address this problem. Unlike conventional mesh refinement strategies based on interpolation error, the proposed approach resolves the convergence and solution jittering issue of the SCP framework by linking the direct and indirect methods with the costate estimation under successive linearization. Along with the development, we introduce a novel costate estimation method and provide proof of its validity. We present numerical simulation results of the proposed method and conduct a comparison with conventional approaches. The results obtained demonstrate successful mitigation of the solution jittering issue and a significant improvement in solution accuracy regarding compliance with the necessary conditions of optimality.

Monochromatic Paths in 2-Edge-Coloured Graphs and Hypergraphs


 
 
 We answer a question of Gyárfás and Sárközy from 2013 by showing that every 2-edge-coloured complete 3-uniform hypergraph can be partitioned into two monochromatic tight paths of different colours. We also give a lower bound for the number of tight paths needed to partition any 2-edge-coloured complete k-partite k-uniform hypergraph. Finally, we show that any 2-edge coloured complete bipartite graph has a partition into a monochromatic cycle and a monochromatic path, of different colours, unless the colouring is a split colouring.
 
 

Lower bound on the size-Ramsey number of tight paths

The size-Ramsey number $R^{(k)}(H)$ of a $k$-uniform hypergraph $H$ is the minimum number of edges in a $k$-uniform hypergraph $G$ with the property that each $2$-edge coloring of $G$ contains a monochromatic copy of $H$. For $k\ge2$ and $n\in\mathbb{N}$, a $k$-uniform tight path on $n$ vertices $P^{(k)}_{n,k-1}$ is defined as a $k$-uniform hypergraph on $n$ vertices for which there is an ordering of its vertices such that the edge set consists of all $k$-element intervals of consecutive vertices in this ordering. We show a lower bound on the size-Ramsey number of $k$-uniform tight paths, which is, considered assymptotically in both the uniformity $k$ and the number of vertices $n$, $R^{(k)}(P^{(k)}_{n,k-1})= \Omega_{n,k}\big(\log (k)n\big)$.

Covering 3‐uniform hypergraphs by vertex‐disjoint tight paths

AbstractFor α > 0 and large integer n, let H be an n‐vertex 3‐uniform hypergraph such that every pair of vertices is in at least edges. We show that contains two vertex‐disjoint tight paths whose union covers the vertex set of . The quantity two here is best possible and the degree condition is asymptotically best possible. This result also has an interpretation as the deficiency problems, recently introduced by Nenadov, Sudakov, and Wagner: every such can be made Hamiltonian by adding at most two vertices and all triples intersecting them.

The Lagrangian density of the disjoint union of a $3$-uniform tight path and a matching and the Turán number of its extension

The Lagrangian density of the disjoint union of a $3$-uniform tight path and a matching and the Turán number of its extension

The size-Ramsey number of 3-uniform tight paths

Given a hypergraph H, the size-Ramsey number r(H) is the smallest integer m such that there exists a graph G with m edges with the property that in any colouring of the edges of G with two colours there is amonochromatic copy of H. We prove that the size-Ramsey number of the 3-uniform tight path on n vertices P_n is linear in n, i.e., r(P_n)=O(n). This answers a question by Dudek, Fleur, Mubayi, and Rödl for 3-uniform hypergraphs [On the size-Ramsey number of hypergraphs, J. Graph Theory 86 (2016), 417-434], who proved r(P_n)=O(n^1.5*log^1.5 n).

Domination numbers and noncover complexes of hypergraphs

Let H be a hypergraph on a finite set V. A cover of H is a set of vertices that meets all edges of H. If W is not a cover of H, then W is said to be a noncover of H. The noncover complex of H is the abstract simplicial complex whose faces are the noncovers of H. In this paper, we study homological properties of noncover complexes of hypergraphs. In particular, we obtain an upper bound on their Leray numbers. The bound is in terms of hypergraph domination numbers. Also, our proof idea is applied to compute the homotopy type of the noncover complexes of certain uniform hypergraphs, called tight paths and tight cycles. This extends to hypergraphs known results on graphs.

Longest Paths in Random Hypergraphs

Given integers $k,j$ with $1\le j \le k-1$, we consider the length of the longest $j$-tight path in the binomial random $k$-uniform hypergraph $H^k(n,p)$. We show that this length undergoes a phase...

Covering and tiling hypergraphs with tight cycles

AbstractA k-uniform tight cycle $C_s^k$ is a hypergraph on s > k vertices with a cyclic ordering such that every k consecutive vertices under this ordering form an edge. The pair (k, s) is admissible if gcd (k, s) = 1 or k / gcd (k,s) is even. We prove that if $s \ge 2{k^2}$ and H is a k-uniform hypergraph with minimum codegree at least (1/2 + o(1))|V(H)|, then every vertex is covered by a copy of $C_s^k$. The bound is asymptotically sharp if (k, s) is admissible. Our main tool allows us to arbitrarily rearrange the order in which a tight path wraps around a complete k-partite k-uniform hypergraph, which may be of independent interest.For hypergraphs F and H, a perfect F-tiling in H is a spanning collection of vertex-disjoint copies of F. For $k \ge 3$, there are currently only a handful of known F-tiling results when F is k-uniform but not k-partite. If s ≢ 0 mod k, then $C_s^k$ is not k-partite. Here we prove an F-tiling result for a family of non-k-partite k-uniform hypergraphs F. Namely, for $s \ge 5{k^2}$, every k-uniform hypergraph H with minimum codegree at least (1/2 + 1/(2s) + o(1))|V(H)| has a perfect $C_s^k$-tiling. Moreover, the bound is asymptotically sharp if k is even and (k, s) is admissible.

The Ramsey Number of Fano Plane Versus Tight Path

The hypergraph Ramsey number of two $3$-uniform hypergraphs $G$ and $H$, denoted by $R(G,H)$, is the least integer~$N$ such that every red-blue edge-coloring of the complete $3$-uniform hypergraph on $N$ vertices contains a red copy of $G$ or a blue copy of $H$. The Fano plane $\mathbb{F}$ is the unique 3-uniform hypergraph with seven edges on seven vertices in which every pair of vertices is contained in a unique edge. There is a simple construction showing that $R(H,\mathbb{F}) \ge 2(v(H)-1) + 1.$ Hypergraphs $H$ for which the equality holds are called $\mathbb{F}$-good. Conlon asked to determine all $H$ that are $\mathbb{F}$-good.In this short paper we make progress on this problem and prove that the tight path of length $n$ is $\mathbb{F}$-good.

Tight paths in convex geometric hypergraphs

One of the most intruguing conjectures in extremal graph theory is the conjecture of Erdős and Sós from 1962, which asserts that every $n$-vertex graph with more than $\frac{k-1}{2}n$ edges contains any $k$-edge tree as a subgraph. Kalai proposed a generalization of this conjecture to hypergraphs. To explain the generalization, we need to define the concept of a tight tree in an $r$-uniform hypergraph, i.e., a hypergraph where each edge contains $r$ vertices. A tight tree is an $r$-uniform hypergraph such that there is an ordering $v_1,\ldots,v_n$ of its its vertices with the following property: the vertices $v_1,\ldots,v_r$ form an edge and for every $i>r$, there is a single edge $e$ containing the vertex $v_i$ and $r-1$ of the vertices $v_1,\ldots,v_{i-1}$, and $e\setminus\{v_i\}$ is a subset of one of the edges consisting only of vertices from $v_1,\ldots,v_{i-1}$. The conjecture of Kalai asserts that every $n$-vertex $r$-uniform hypergraph with more than $\frac{k-1}{r}\binom{n}{r-1}$ edges contains every $k$-edge tight tree as a subhypergraph. The recent breakthrough results on the existence of combinatorial designs by Keevash and by Glock, Kühn, Lo and Osthus show that this conjecture, if true, would be tight for infinitely many values of $n$ for every $r$ and $k$. The article deals with the special case of the conjecture when the sought tight tree is a path, i.e., the edges are the $r$-tuples of consecutive vertices in the above ordering. The case $r=2$ is the famous Erdős-Gallai theorem on the existence of paths in graphs. The case $r=3$ and $k=4$ follows from an earlier work of the authors on the conjecture of Kalai. The main result of the article is the first non-trivial upper bound valid for all $r$ and $k$. The proof is based on techniques developed for a closely related problem where a hypergraph comes with a geometric structure: the vertices are points in the plane in a strictly convex position and the sought path has to zigzag beetwen the vertices.

Lagrangian densities of linear forests and Turán numbers of their extensions

AbstractThe Lagrangian of a hypergraph has been a useful tool in hypergraph extremal problems. Recently, Lagrangian densities of hypergraphs and Turán numbers of their extensions have been studied actively. However, determining the Lagrangian density of a hypergraph is not an easy task even for a “simple” hypergraph. For example, to determine the Lagrangian density of is equivalent to determine the Turán density of (a long standing conjecture of Turán). Hefetz and Keevash studied the Lagrangian density of the 3‐uniform matching of size 2. Pikhurko determined the Lagrangian density of a 4‐uniform tight path of length 2 and this led to confirm the conjecture of Frankl and Füredi on the Turán number of the ‐uniform generalized triangle for the case . It is natural and interesting to consider Lagrangian densities of other “basic” hypergraphs. In this paper, we determine the Lagrangian densities for a class of 3‐uniform linear forests. For positive integers and , let be the disjoint union of a 3‐uniform linear path of length and pairwise disjoint edges. In this paper, we determine the Lagrangian densities of for any and or 3. Applying a modified version of Pikhurko's transference argument used by Brandt, Irwin, and Jiang, we obtain the Turán numbers of their extensions.

Hypergraphs Not Containing a Tight Tree with a Bounded Trunk

An $r$-uniform hypergraph is a tight $r$-tree if its edges can be ordered so that every edge $e$ contains a vertex $v$ that does not belong to any preceding edge and the set $e-v$ lies in some preceding edge. A conjecture of Kalai personal communication published in Frankl and Füredi, J. Combin. Theory Ser. A, 45 (1987), pp. 226--262, generalizing the Erdös--Sós conjecture for trees, asserts that if $T$ is a tight $r$-tree with $t$ edges and $G$ is an $n$-vertex $r$-uniform hypergraph containing no copy of $T$, then $G$ has at most $\frac{t-1}{r}\binom{n}{r-1}$ edges. A trunk $T'$ of a tight $r$-tree $T$ is a tight subtree such that every edge of $T-T'$ has $r-1$ vertices in some edge of $T'$ and a vertex outside $T'$. For $r\ge 3$, the only nontrivial family of tight $r$-trees for which this conjecture has been proved is the family of $r$-trees with trunk size one in J. Combin. Theory Ser. A, 45 (1987), pp. 226--262. Our main result is an asymptotic version of Kalai's conjecture for all tight trees $T$ of bounded trunk size. This follows from our upper bound on the size of a $T$-free $r$-uniform hypergraph $G$ in terms of the size of its shadow. We also give a short proof of Kalai's conjecture for tight $r$-trees with at most four edges. In particular, for 3-uniform hypergraphs, our result on the tight path of length $4$ implies the intersection shadow theorem of Katona Acta Math. Acad. Sci. Hungar., 15 (1964), pp. 329--337.

On extremal hypergraphs for forests of tight paths

In this paper, we investigate the maximal size of a k-uniform hypergraph containing no forests of tight paths, which extends the classical Erdős–Gallai Theorem for paths in graphs. Our results build on the results of Györi, Katona and Lemons, who considered the maximal size of a k-uniform hypergraph containing no single tight path.

On the Size-Ramsey Number of Tight Paths

For any $r\geq 2$ and $k\geq 3$, the $r$-color size-Ramsey number $\hat R(\mathcal{G},r)$ of a $k$-uniform hypergraph $\mathcal{G}$ is the smallest integer $m$ such that there exists a $k$-uniform ...

Learning graphs and 1-uniform dcsl graphs

A distance compatible set labeling (dcsl) of a connected graph [Formula: see text] is an injective set assignment [Formula: see text] [Formula: see text] being a non-empty ground set, such that the corresponding induced function [Formula: see text] given by [Formula: see text] satisfies [Formula: see text] for every pair of distinct vertices [Formula: see text] where [Formula: see text] denotes the path distance between [Formula: see text] and [Formula: see text] and [Formula: see text] is a constant, not necessarily an integer, depending on the pair of vertices [Formula: see text] chosen. A dcsl [Formula: see text] of [Formula: see text] is [Formula: see text]-uniform if all the constants of proportionality with respect to [Formula: see text] are equal to [Formula: see text] and if [Formula: see text] admits such a dcsl then [Formula: see text] is called a [Formula: see text]-uniform dcsl graph. The family [Formula: see text] is well-graded family, if there is a tight path between any two of its distinct sets. A learning graph is an [Formula: see text]-induced graph of a learning space. In this paper, we initiate a study on subgraphs of 1-uniform graphs which lead to the study of Knowledge Structures, Learning Spaces and Union-closed conjecture using graph theory techniques.

Off-diagonal hypergraph Ramsey numbers

The Ramsey number rk(s,n) is the minimum N such that every red–blue coloring of the k-subsets of {1,…,N} contains a red set of size s or a blue set of size n, where a set is red (blue) if all of its k-subsets are red (blue). Let rk(Ps,n) be the minimum N such that every red–blue coloring of the k-subsets of {1,…,N} results in a red ordered tight path with s vertices or a blue set of size n. The problem of estimating both rk(s,n) and rk(Ps,n) for k=2 goes back to the seminal work of Erdős and Szekeres from 1935, while the case k≥3 was first investigated by Erdős and Rado in 1952.In this paper, we deduce the first quantitative relationship between multicolor variants of rk(Ps,n) and rk(n,n). This yields several consequences including the following:•We determine the correct tower growth rate for both rk(s,n) and rk(Ps,n) for s≥k+3. The question of determining the tower growth rate of rk(s,n) for all s≥k+1 was posed by Erdős and Hajnal in 1972 and this is almost settled here.•We show that determining the tower growth rate of r4(P5,n) is equivalent to determining the tower growth rate of rk(n,n) for all k≥3, which is a notorious conjecture of Erdős, Hajnal and Rado from 1965 that remains open.

Variants of the Erdős–Szekeres and Erdős–Hajnal Ramsey problems

Given integers ℓ,n, the ℓth power of the path Pn is the ordered graph Pnℓ with vertex set v1<v2<⋯<vn and all edges of the form vivj where |i−j|≤ℓ. The Ramsey number r(Pnℓ,Pnℓ) is the minimum N such that every 2-coloring of [N]2 results in a monochromatic copy of Pnℓ. It is well-known that r(Pn1,Pn1)=(n−1)2+1. For ℓ>1, Balko–Cibulka–Král–Kynčl proved that r(Pnℓ,Pnℓ)<cℓn128ℓ and asked for the growth rate for fixed ℓ. When ℓ=2, we improve this upper bound substantially by proving r(Pn2,Pn2)<cn19.5. Using this result, we determine the correct tower growth rate of the k-uniform hypergraph Ramsey number of a (k+1)-clique versus an ordered tight path. Finally, we consider an ordered version of the classical Erdős–Hajnal hypergraph Ramsey problem, improve the tower height given by the trivial upper bound, and conjecture that this tower height is optimal.

Constructions of 1-Uniform dcsl graphs using Well-graded families of sets

A 1-uniform dcsl of a graph G is an injective set assignment function f:V(G)→2X, X be a non-empty set, such that the corresponding induced function f⊕:V(G)×V(G)→2X\\{ϕ} given by f⊕(uv)=f(u)⊕f(v) satisfies |f⊕(u,v)|=1.d(u,v) for all distinct u,v∈V(G), where d(u,v) is the length of a shortest path between u and v, and f(u)⊕f(v) denotes the symmetric difference of the two sets. Let F be a family of subsets of a set X. A tight path between two distinct sets P and Q (or from P to Q) in F is a sequence P0=P,P1,P2…Pn=Q in F such that d(P,Q)=|PΔQ|=n and d(Pi,Pi+1)=1 for 0≤i≤n−1. The family F is well-graded (or wg-family), if there is a tight path between any two of its distinct sets. Any family F of subsets of X defines a graph GF=(F,EF), where EF={{P,Q}⊆F:|PΔQ|=1}, and we call GF, anF-induced graph. The purpose of this paper is to examine the existence of 1-uniform dcsl of an induced graph GF1∪F2∪…Fn formed from the finite union of well-graded families F1,F2,…, and Fn by introducing amalgamation techniques in between them, where, for 1≤i≤n, each Fi-induced graph, GFi is isomorphic to a 1-uniform dcsl even cycle.

Characterization of Vertex labeling of 1-uniform dcsl graph which form a lattice

A 1-uniform dcsl of a graph $G$ is an injective set-valued function $f : V(G)\rightarrow 2^{X}, X$ be a non-empty ground set, such that the corresponding induced function $f^{\oplus} :V(G) \times V(G) \rightarrow 2^{X}\setminus \{\phi\}$ satisfies $\vert f^{\oplus}(u, v)\vert = 1.d(u, v)$ for all distinct $u, v \in V(G)$, where $d(u, v)$ is the distance between $u$ and $v$. Let ${\mathscr{F}}$ be a family of subsets of a set $X$. A {\it tight path} between two distinct sets $P$ and $Q$ (or from $P$ to $Q$) in ${\mathscr{F}}$ is a sequence $P_0 = P, P_1, P_2 \dots P_n = Q$ in ${\mathscr{F}}$ such that $d(P,Q)= \mid P \bigtriangleup Q \mid = n$ and $d(P_i, P_{i+1}) = 1$ for $0 \leq i\leq n-1$. The family ${\mathscr{F}}$ is {\it well-graded} (or {\it wg-family}), if there is a {\it tight path} between any two of its distinct sets. Any family ${\mathscr{F}}$ of subsets of $X$ defines a graph $G_{\mathscr{F}} = ( {\mathscr{F}} , E_{\mathscr{F}})$, where $E_{\mathscr{F}} = \{ \{P,Q \} \subseteq {\mathscr{F}}:\mid P \bigtriangleup Q \mid = 1 \} $, and we call $G_{\mathscr{F}$, an ${\mathscr{F}}$-induced graph. In this paper, we study 1-uniform dcsl graphs whose vertex labelings whether or not forms a lattice and prove that the cover graph $C_{{\mathscr{F}}}$ of a poset ${\mathscr{F}}$ with respect to set inclusion ` $\subseteq$' is isomorphic to the ${\mathscr{F}}$-induced graph $G_{{\mathscr{F}}}$.