The length polyhedron of an interval order P is the convex hull of integer vectors representing the interval lengths in possible interval representations of P in which all intervals have integer endpoints. This polyhedron is an integral translation of a polyhedral cone, with its apex corresponding to the canonical interval representation of P (also known as the minimal endpoint representation). In earlier work, we introduced an arc-weighted directed graph model, termed the key graph, inspired by this canonical representation. We showed that cycles in the key graph correspond, via Fourier-Motzkin elimination, to inequalities that describe supporting hyperplanes of the length polyhedron. These cycle inequalities derived from the key graph form a complete system of linear inequalities defining the length polyhedron. By applying a theorem due to Cook, we establish here that this system of inequalities is totally dual integral (TDI). Leveraging circulations, total dual integrality, and the special structure of the key graph, our main theorem demonstrates that a cycle inequality is a positive linear combination of other cycle inequalities if and only if it is a positive integral linear combination of smaller cycle inequalities (where 'smaller' here refers a natural weak ordering among these cycle inequalities). This yields an efficient method to remove redundant cycle inequalities and ultimately construct the unique minimal TDI-system, also known as the Schrijver system, for the length polyhedron. Notably, if the key graph contains a polynomial number of cycles, this gives a polynomial-time algorithm to compute the Schrijver system for the length polyhedron. Lastly, we provide examples of interval orders where the Schrijver system has an exponential size. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

The length polyhedron QP of an interval order P is the convex hull of integral vectors representing the interval lengths in interval representations of P. This polyhedron has been studied by various authors, including Fishburn and Isaak. Notably, QP forms a pointed affine cone, a property inherited from being a projection of the representation polyhedron, a structure explored also by Doignon and Pauwels. The apex of the length polyhedron corresponds to Greenough’s minimal endpoint representation, which is, in fact, the length vector of the canonical interval representation—an interval representation that minimizes the sum of the interval lengths.Building on a combinatorial perspective of canonical representations, we refine Isaak’s graph-theoretical model by introducing a new and simpler directed graph. From directed cycles of this key digraph, we extract a linear system of inequalities that precisely characterizes the length polyhedron QP. This combinatorial approach also reveals the unique Hilbert basis of the polyhedron, which consists of binary rays. We prove that the intersection graph of the sets corresponding to these binary rays are Berge graphs; therefore they are perfect graphs. As a result, for interval orders with bounded width, the length polyhedron has a polynomial-sized Hilbert basis, which can be computed in polynomial time. We also provide an example of a family of interval orders with a Hilbert basis of exponential size. In a companion paper we determine the Schrijver system for the length polyhedron. We conclude with open problems.

The interval count problem, a classical question in the study of interval orders, was introduced by Ronald Graham in the 1980s. This problem asks: given an interval order P, what is the minimum number of distinct interval lengths required to construct an interval representation of P? Interval orders that can be represented with just one interval length are known as semiorders, and their characterization is well known. However, the characterization of interval orders that require at most k interval lengths — termed k-count interval orders—remains an open and challenging problem for k≥ 2 . Our investigation into 2-count interval orders led us naturally to consider a related problem, interval representations of permutations, which we introduce in this paper. Specifically, we characterize permutations that have a 2-count interval representation. We prove that a permutation admits a 2-count interval representation if and only if its longest decreasing subsequences have length at most 2. For larger values of k, however, a similar characterization does not hold. There are permutations that do not permit a 3-count interval representation even though all of their decreasing subsequences have length at most 3. Characterizing k-count permutations remains open for k ≥ 3 . The k-count permutation representation problem appears to capture essential aspects of the broader problem of characterizing k-count interval orders. To support this connection, we apply our findings on interval representations of permutations to demonstrate that a height-3 interval order is 2-count if and only if it has depth at most 2, where the depth of an interval order is the minimum, over all interval representations of the order, of the maximum number of intervals in a chain of nested intervals.

Thelong-standingSzemerédiandPetruskaconjecture(StudiaSciMathHungar7:363– 374,1972)wasrecentlyresolvedasymptotically(KézdyandLehelinDiscreteMath 346:113469,2023).SeveraldecadesagoGyárfásetal.(JCombinTheorySerB 33:161–165,1982)observed,viaastraightforwardbutunpublishedargument,that thisconjectureisequivalenttotheproblemofdeterminingthemaximumorderofa 3-uniform τ -critical hypergraph. Consequently, an asymptotically tight upper bound for the maximum order of a 3-uniform τ -critical hypergraph follows from our recent work, reawakening interest in this equivalence. In this companion paper we supply a simple proof of this equivalence. We also present related background with open problems, and mention combinatorial geometry applications of the Szemerédi and Petruska conjecture.

The long-standing Szemerédi and Petruska conjecture (Studia Sci Math Hungar 7:363–374, 1972) was recently resolved asymptotically (Kézdy and Lehel in Discrete Math 346:113469, 2023). Several decades ago Gyárfás et al. (J Combin Theory Ser B 33:161–165, 1982) observed, via a straightforward but unpublished argument, that this conjecture is equivalent to the problem of determining the maximum order of a 3-uniform τ -critical hypergraph. Consequently, an asymptotically tight upper bound for the maximum order of a 3-uniform τ -critical hypergraph follows from our recent work, reawakening interest in this equivalence. In this companion paper we supply a simple proof of this equivalence. We also present related background with open problems, and mention combinatorial geometry applications of the Szemerédi and Petruska conjecture.

Consider a $3$-uniform hypergraph of order $n$ with clique number $k$ such that the intersection of all its $k$-cliques is empty. Szemer\'edi and Petruska proved $n\leq 8m^2+3m$, for fixed $m=n-k$, and they conjectured the sharp bound $n \leq {m+2 \choose 2}$. This problem is known to be equivalent to determining the maximum order of a $\tau$-critical $3$-uniform hypergraph with transversal number $m$ (details may also be found in a companion paper: arXiv:2204.02859). The best known bound, $n\leq \frac{3}{4}m^2+m+1$, was obtained by Tuza using the machinery of $\tau$-critical hypergraphs. Here we propose an alternative approach, a combination of the iterative decomposition process introduced by Szemer\'edi and Petruska with the skew version of Bollob\'as's theorem on set pair systems. The new approach improves the bound to $n\leq {m+2 \choose 2} + O(m^{{5}/{3}})$, resolving the conjecture asymptotically.

Two sets overlap if they intersect and neither contains the other. Given a family $$\mathcal {F}$$ of sets, a system of overlap representation (SOR) for $$\mathcal {F}$$ assigns to each set $$X\in \mathcal {F}$$ a subset $$S_X$$ of X, called its representative set, so that the representative sets chosen for any two overlapping members of $$\mathcal {F}$$ intersect. Let $$\mathcal {F}_n$$ be the family of intervals of integers contained in $$\{1,\ldots ,n\}$$ , and let f(n) be the minimum of the maximum size of the sets in an SOR of $$\mathcal {F}_n$$ . We prove $$(1-o(1))\lg (n-1)<f(n)\le 2\lg (n-1).$$

A graph is an apex graph if it contains a vertex whose deletion leaves a planar graph. The family of apex graphs is minor-closed and so it is characterized by a finite list of minor-minimal non-members. The long-standing problem of determining this finite list of apex obstructions remains open. This paper determines the $133$ minor-minimal, non-apex graphs that have connectivity two.

A derivation (of order 1) satisfies the reduction formula $$f(x^k) = kx^{k-1}f(x)$$ for any integer k. In this article we find corresponding reduction formulas for derivations of higher order on commutative rings. More precisely, for every derivation f of order n and every positive integer k we find an explicit formula for $$f(x^k)$$ as a linear combination of $$x^{k-1}f(x),x^{k-2}f(x^2), \ldots , x^{k-n}f(x^n)$$ . The proof hinges on the hypergeometric identity $$\begin{aligned} \sum _{k \ge 0} (-1)^k \left( {\begin{array}{c}n\\ k\end{array}}\right) \left( {\begin{array}{c}n+d+1-k\\ n-j\end{array}}\right) \left( {\begin{array}{c}d+j-k\\ j\end{array}}\right) = \left( {\begin{array}{c}n\\ j\end{array}}\right) \end{aligned}$$ for any positive integer n, nonnegative integer d, and integer j satisfying $$0 \le j \le n$$ . We prove this identity via the WZ-method.

For a given integer $$k\ge 1$$ , a graph G with at least 2k vertices is called k-path-pairable, if for any set of k disjoint pairs of vertices, $$s_i,t_i$$ , $$1\le i\le k$$ , there exist pairwise edge-disjoint $$s_i,t_i$$ -paths in G. The path-pairability numberis the largest k such that G is k-path-pairable. Bounds on the path-pairability number are given here if G is the graph of infinite integer grids in the Euclidean plane. We prove that the path-pairability number of the integer quadrant is 4, and we show that the integer half-plane is 6-path-pairable and at most 7-path-pairable.

Let H be a 3-uniform hypergraph of order n with clique number $$\omega (H)=k$$ . Assume that the union of the k-cliques of H equals its vertex set, the intersection of all maximum cliques of H is empty, but the intersection of all but one k-clique is non-empty. For fixed $$m=n-k$$ , Szemerédi and Petruska conjectured the sharp bound $$n\hbox {\,\,\char 054\,\,}{m+2\atopwithdelims ()2}$$ . In this note the conjecture is verified for $$m=2,3$$ and 4.

Eckhoff proposed a combinatorial version of the classical Hadwiger–Debrunner $(p,q)$-problems as follows. Let ${\cal F}$ be a finite family of convex sets in the plane and let $m\geqslant 1$ be an integer. If among every ${m+2\choose 2}$ members of ${\cal F}$ all but at most $m-1$ members have a common point, then there is a common point for all but at most $m-1$ members of ${\cal F}$. The claim is an extension of Helly's theorem ($m=1$). The case $m=2$ was verified by Nadler and by Perles. Here we show that Eckhoff 's conjecture follows from an old conjecture due to Szemerédi and Petruska concerning $3$-uniform hypergraphs. This conjecture is still open in general; its solution for a few special cases answers Eckhoff's problem for $m=3,4$. A new proof for the case $m=2$ is also presented.

If the bisection width of a 2-connected graph G of even order n is not less than n/2, i.e., if the graph satisfies the even cut condition, then G has at least 3n/2 - 2 edges. Here we characterize the 2-connected extremal graphs with 3n/2 - 2 edges for every n >= 4. For n >= 10 an extremal graph has two high-degree vertices, the other vertices have degree 2 and they are located on paths suspended between these poles. (C) 2020 Elsevier B.V. All rights reserved.

The bisection width is the minimum number of edges required to split the vertex set of a graph into two (nearly) equal parts. Monien and Preis proved that the bisection width of a cubic graph with n nodes is bounded above by n∕6+o(n). Here we show that every cubic graph of even order n≥16 has bisection width less than n∕2, thus these graphs violate the even cut condition (ECC). All edge-minimal subcubic graphs satisfying ECC are also described. The bisection width is a reference parameter to compare networks for parallel architectures; ECC is a property necessary for bottleneck free all-to-all communications.

Given 2k−1 convex sets in R2 such that no point of the plane is covered by more than k of the sets, is it true that there are two among the convex sets whose union contains all k-covered points of the plane? This question due to Gy. Petruska has an obvious affirmative answer for k=1,2,3; we show here that the claim is also true for k=4, and we present a counterexample for k=5. We explain how Petruska’s geometry question fits into the classical hypergraph extremal problems, called arrow problems, proposed by P. Erdős.

The study of a graph theory model of certain telecommunications network problems lead to the concept of path-pairability, a variation of weak linkedness of graphs. A graph G is k-path-pairable if for any set of 2k distinct vertices, si, ti, 1 ≤ i ≤ k, there exist pairwise edge-disjoint si, ti-paths in G, for 1 ≤ i ≤ k. The path-pairability number is the largest k such that G is k-path-pairable. Cliques, stars, the Cartesian product of two cliques (of order at least three) are ‘fully pairable’; that is ⌊n/2⌋-pairable, where n is the order of the graph. Here we determine the path-pairability number of the Cartesian product of two stars.

Consider a 3-uniform hypergraph of order n with clique number k such that the intersection of all its k-cliques is empty. Szemerédi and Petruska proved n≤ 8m^2+3m, for fixed m=n-k, and they conjectured the sharp bound n≤m+2 2. Tuza proved the best known bound, n≤3/4m^2+m+1, using the machinery of τ-critical hypergraphs. Here we propose an alternative approach, combining a decomposition process introduced by Szemerédi and Petruska with the skew version of Bollobás's theorem to prove n≤ m^2 + 6m + 2. While the bound obtained here is weaker than Tuza's bound, it is a proof-of-concept for a different approach and a call to apply dimension bounds from linear algebra.

For k fixed, a graph G is k-path-pairable, if for any set of k disjoint pairs of vertices, si,ti, 1≤i≤k, there exist pairwise edge-disjoint si,ti-paths in G. Bounds on path-pairability are given here if G is the graph of the infinite integer grid in the Euclidean plane (vertices of G are the points of integer coordinates and two vertices are adjacent if and only if their Manhattan distance is 1). We prove that G is 10-path-pairable and at most 14-path-pairable. Related results and conjectures are summarized also for the integer halfplane, for the positive integer quadrant and for finite grids.

Let G=P_6 P_6 be the 6× 6 grid, the Cartesian product of two paths of six vertices. Let T be the set of eight distinct vertices of G, called terminals, and assume that T is partitioned into four terminal pairs {s_i,t_i}, 1≤ i≤ 4. We prove that G is 4-path-pairable, that is, for every T there exist in G pairwise edge disjoint s_i,t_i-paths, 1≤ i≤ 4.

Let $Q$ be a finite subgraph of the integer grid $G$ in the plane, and let $T$ be a set of pairs of distinct vertices in $G$, called `terminal pairs'. Escaping a subset $X\subset T\cap Q$ from $Q$ means finding edge disjoint paths from the terminals in $X$ into distinct vertices of a set $L$ in the boundary of $Q$. Here we prove several lemmas for the cases where $Q$ is a $3\times 3$ grid, $L$ is the union of a vertical and horizontal boundary line of $Q$, furthermore, $T$ is a set of four terminal pairs in $G$ such that $|T\cap Q|\geq 5$. These lemmas are applied in [4] and complete the proof that the Cartesian product of two (one way) infinite paths has path-pairability number four.