Combinatorica最新文献

We present problems and results that combine graph-minors and coarse geometry. For example, we ask whether every geodesic metric space (or graph) without a fat H minor is quasi-isometric to a graph with no H minor, for an arbitrary finite graph H. We answer this affirmatively for a few small H. We also present a metric analogue of Menger’s theorem and König’s ray theorem. We conjecture metric analogues of the Erdős–Pósa Theorem and Halin’s grid theorem.

In a digraph, a dicut is a cut where all the arcs cross in one direction. A dijoin is a subset of arcs that intersects each dicut. Woodall conjectured in 1976 that in every digraph, the minimum size of a dicut equals to the maximum number of disjoint dijoins. However, prior to our work, it was not even known whether at least 3 disjoint dijoins exist in an arbitrary digraph whose minimum dicut size is sufficiently large. By building connections with nowhere-zero (circular) k-flows, we prove that every digraph with minimum dicut size (tau ) contains (leftlfloor frac{tau }{k}rightrfloor ) disjoint dijoins if the underlying undirected graph admits a nowhere-zero (circular) k-flow. The existence of nowhere-zero 6-flows in 2-edge-connected graphs (Seymour 1981) directly leads to the existence of (leftlfloor frac{tau }{6}rightrfloor ) disjoint dijoins in a digraph with minimum dicut size (tau ), which can be found in polynomial time as well. The existence of nowhere-zero circular (frac{2p+1}{p})-flows in 6p-edge-connected graphs (Lovász et al. 2013) directly leads to the existence of (leftlfloor frac{tau p}{2p+1}rightrfloor ) disjoint dijoins in a digraph with minimum dicut size (tau ) whose underlying undirected graph is 6p-edge-connected. We also discuss reformulations of Woodall’s conjecture into packing strongly connected orientations.

For a finite abelian group G and a positive integer k, let (textsf{D}_k(G)) denote the smallest integer (ell ) such that each sequence over G of length at least (ell ) has k disjoint nontrivial zero-sum subsequences. It is known that (mathsf D_k(G)=n_1+kn_2-1) if (Gcong C_{n_1}oplus C_{n_2}) is a rank 2 group, where (1<n_1, | ,n_2). We investigate the associated inverse problem for rank 2 groups, that is, characterizing the structure of zero-sum sequences of length (mathsf D_k(G)) that can not be partitioned into (k+1) nontrivial zero-sum subsequences.

A long-standing open conjecture of Branko Grünbaum from 1972 states that any simple arrangement of n pairwise intersecting pseudocircles in the plane can have at most (2n-2) digons. Agarwal et al. proved this conjecture for arrangements of pairwise intersecting pseudocircles in which there is a common point surrounded by all pseudocircles. Recently, Felsner, Roch and Scheucher showed that Grünbaum’s conjecture is true for arrangements of pairwise intersecting pseudocircles in which there are three pseudocircles every pair of which create a digon. In this paper we prove this over 50-year-old conjecture of Grünbaum for any simple arrangement of pairwise intersecting circles in the plane.

Suppose that k is a function of n and . We show that with probability (1-O(1/n)), a uniformly random (ktimes n) Latin rectangle contains no proper Latin subsquare of order 4 or more, proving a conjecture of Divoux, Kelly, Kennedy and Sidhu. We also show that the expected number of subsquares of order 3 is bounded and find that the expected number of subsquares of order 2 is (left( {begin{array}{c}k 2end{array}}right) (1/2+o(1))) for all (kleqslant n).

The Liebeck–Nikolov–Shalev conjecture (Bull Lond Math Soc 44(3):469–472, 2012) asserts that, for any finite simple non-abelian group G and any set (Asubseteq G) with (|A|ge 2), G is the product of at most (Nfrac{log |G|}{log |A|}) conjugates of A, for some absolute constant N. For G of Lie type, we prove that for any (varepsilon >0) there is some (N_{varepsilon }) for which G is the product of at most (N_{varepsilon }left( frac{log |G|}{log |A|}right) ^{1+varepsilon }) conjugates of either A or (A^{-1}). For symmetric sets, this improves on results of Liebeck et al. (2012) and Gill et al. (Groups Geom Dyn 7(4):867–882, 2013). During the preparation of this paper, the proof of the Liebeck–Nikolov–Shalev conjecture was completed by Lifshitz (Completing the proof of the Liebeck–Nikolov–Shalev conjecture, 2024, https://arxiv.org/abs/2408.10127). Both papers use Gill et al. (Initiating the proof of the Liebeck–Nikolov–Shalev conjecture, 2024, https://arxiv.org/abs/2408.07800) as a starting point. Lifshitz’s argument uses heavy machinery from representation theory to complete the conjecture, whereas this paper achieves a more modest result by rather elementary combinatorial arguments.

We obtain new results on the Turán number of any bounded degree uniform hypergraph obtained as the expansion of a hypergraph of bounded uniformity. These are asymptotically sharp over an essentially optimal regime for both the uniformity and the number of edges and solve a number of open problems in Extremal Combinatorics. Firstly, we give general conditions under which the crosscut parameter asymptotically determines the Turán number, thus answering a question of Mubayi and Verstraëte. Secondly, we refine our asymptotic results to obtain several exact results, including proofs of the Huang–Loh–Sudakov conjecture on cross matchings and the Füredi–Jiang–Seiver conjecture on path expansions. We have introduced two major new tools for the proofs of these results. The first of these, Global Hypercontractivity, is used as a ‘black box’ (we present it in a separate paper with several other applications). The second tool, presented in this paper, is a far-reaching extension of the Junta Method, which we develop from a powerful and general technique for finding matchings in hypergraphs under certain pseudorandomness conditions.

Given an n-element set (Csubseteq mathbb {R}^d) and a (sufficiently generic) k-element multiset (Vsubseteq mathbb {R}^d), we can order the points in C by ranking each point (cin C) according to the sum of the distances from c to the points of V. Let (Psi _k(C)) denote the set of orderings of C that can be obtained in this manner as V varies, and let (psi ^{textrm{max}}_{d,k}(n)) be the maximum of (|Psi _k(C)|) as C ranges over all n-element subsets of (mathbb {R}^d). We prove that (psi ^{textrm{max}}_{d,k}(n)=Theta _{d,k}(n^{2dk})) when (d ge 2) and that (psi ^{textrm{max}}_{1,k}(n)=Theta _k(n^{4lceil k/2rceil -2})). As a step toward proving this result, we establish a bound on the number of sign patterns determined by a collection of functions that are sums of radicals of nonnegative polynomials; this can be understood as an analogue of a classical theorem of Warren. We also prove several results about the set (Psi (C)=bigcup _{kge 1}Psi _k(C)); this includes an exact description of (Psi (C)) when (d=1) and when C is the set of vertices of a vertex-transitive polytope.

A subset S of real numbers is called bi-Sidon if it is a Sidon set with respect to both addition and multiplication, i.e., if all pairwise sums and all pairwise products of elements of S are distinct. Imre Ruzsa asked the following question: What is the maximum number f(N) such that every set S of N real numbers contains a bi-Sidon subset of size at least f(N)? He proved that (f(N)geqslant cN^{frac{1}{3}}), for a constant (c>0). In this note, we improve this bound to (N^{frac{1}{3}+frac{7}{78}+o(1)}).