Combinatorica最新文献

We show that, for any prime p and integer (k ge 2), a simple ({{,textrm{GF},}}(p))-representable matroid with sufficiently high rank has a rank-k flat which is either independent in M, or is a projective or affine geometry. As a corollary we obtain a Ramsey-type theorem for ({{,textrm{GF},}}(p))-representable matroids. For any prime p and integer (kge 2), if we 2-colour the elements in any simple ({{,textrm{GF},}}(p))-representable matroid with sufficiently high rank, then there is a monochromatic flat of rank k.

For any integer (hgeqslant 2), a set of integers (B={b_i}_{iin I}) is a (B_h)-set if all h-sums (b_{i_1}+ldots +b_{i_h}) with (i_1<ldots <i_h) are distinct. Answering a question of Alon and Erdős [2], for every (hgeqslant 2) we construct a set of integers X which is not a union of finitely many (B_h)-sets, yet any finite subset (Ysubseteq X) contains an (B_h)-set Z with (|Z|geqslant varepsilon |Y|), where (varepsilon :=varepsilon (h)). We also discuss questions related to a problem of Pisier about the existence of a set A with similar properties when replacing (B_h)-sets by the requirement that all finite sums (sum _{jin J}b_j) are distinct.

Let G be a 3-connected ordered graph with n vertices and m edges. Let (textbf{p}) be a randomly chosen mapping of these n vertices to the integer range ({1, 2,3, ldots , 2^b}) for (bge m^2). Let (ell ) be the vector of m Euclidean lengths of G’s edges under (textbf{p}). In this paper, we show that, with high probability over (textbf{p}), we can efficiently reconstruct both G and (textbf{p}) from (ell ). This reconstruction problem is NP-HARD in the worst case, even if both G and (ell ) are given. We also show that our results stand in the presence of small amounts of error in (ell ), and in the real setting, with sufficiently accurate length measurements. Our method combines lattice reduction, which has previously been used to solve random subset sum problems, with an algorithm of Seymour that can efficiently reconstruct an ordered graph given an independence oracle for its matroid.

A directed diameter of a directed graph is the maximum possible distance between a pair of vertices, where paths must respect edge orientations, while undirected diameter is the diameter of the undirected graph obtained by symmetrizing the edges. In 2006 Babai proved that for a connected directed Cayley graph on ( n ) vertices the directed diameter is bounded above by a polynomial in undirected diameter and ( log n ). Moreover, Babai conjectured that a similar bound holds for vertex-transitive graphs. We prove this conjecture of Babai, in fact, it follows from a more general bound for connected relations of homogeneous coherent configurations. The main novelty of the proof is a generalization of Ruzsa’s triangle inequality from additive combinatorics to the setting of graphs

An amply regular graph is a regular graph such that any two adjacent vertices have (alpha ) common neighbors and any two vertices with distance 2 have (beta ) common neighbors. We prove a sharp lower bound estimate for the Lin–Lu–Yau curvature of any amply regular graph with girth 3 and (beta >alpha ). The proof involves new ideas relating discrete Ricci curvature with local matching properties: This includes a novel construction of a regular bipartite graph from the local structure and related distance estimates. As a consequence, we obtain sharp diameter and eigenvalue bounds for amply regular graphs.

In 1977 Diaconis and Graham proved two inequalities relating different measures of disarray in permutations, and asked for a characterization of those permutations for which equality holds in one of these inequalities. Such a characterization was first given in 2013. Recently, another characterization was given by Woo, using a topological link in ({mathbb {R}}^3) that can be associated to the cycle diagram of a permutation. We show that Woo’s characterization extends much further: for any permutation, the discrepancy in Diaconis and Graham’s inequality is directly related to the Euler characteristic of the associated link. This connection provides a new proof of the original result of Diaconis and Graham. We also characterize permutations with a fixed discrepancy in terms of their associated links and find that the stabilized-interval-free permutations are precisely those whose associated links are nonsplit.

Reidl et al. (Eur J Comb 75:152–168, 2019) characterized graph classes of bounded expansion as follows: A class ({mathcal {C}}) closed under subgraphs has bounded expansion if and only if there exists a function (f:{mathbb {N}} rightarrow {mathbb {N}}) such that for every graph (G in {mathcal {C}}), every nonempty subset A of vertices in G and every nonnegative integer r, the number of distinct intersections between A and a ball of radius r in G is at most f(r) |A|. When ({mathcal {C}}) has bounded expansion, the function f(r) coming from existing proofs is typically exponential. In the special case of planar graphs, it was conjectured by Sokołowski (Electron J Comb 30(2):P2.3, 2023) that f(r) could be taken to be a polynomial. In this paper, we prove this conjecture: For every nonempty subset A of vertices in a planar graph G and every nonnegative integer r, the number of distinct intersections between A and a ball of radius r in G is ({{,mathrm{{mathcal {O}}},}}(r^4 |A|)). We also show that a polynomial bound holds more generally for every proper minor-closed class of graphs.

Given a graph G with a set F(v) of forbidden values at each (v in V(G)), an F-avoiding orientation of G is an orientation in which (deg ^+(v) not in F(v)) for each vertex v. Akbari, Dalirrooyfard, Ehsani, Ozeki, and Sherkati conjectured that if (|F(v)| < frac{1}{2} deg (v)) for each (v in V(G)), then G has an F-avoiding orientation, and they showed that this statement is true when (frac{1}{2}) is replaced by (frac{1}{4}). In this paper, we take a step toward this conjecture by proving that if (|F(v)| < lfloor frac{1}{3} deg (v) rfloor ) for each vertex v, then G has an F-avoiding orientation. Furthermore, we show that if the maximum degree of G is subexponential in terms of the minimum degree, then this coefficient of (frac{1}{3}) can be increased to (sqrt{2} - 1 - o(1) approx 0.414). Our main tool is a new sufficient condition for the existence of an F-avoiding orientation based on the Combinatorial Nullstellensatz of Alon and Tarsi.

Let (D=(V,A)) be a digraph. For an integer (kge 1), a k-arc-connected flip is an arc subset of D such that after reversing the arcs in it the digraph becomes (strongly) k-arc-connected. The first main result of this paper introduces a sufficient condition for the existence of a k-arc-connected flip that is also a submodular flow for a crossing submodular function. More specifically, given some integer (tau ge 1), suppose (d_A^+(U)+(frac{tau }{k}-1)d_A^-(U)ge tau ) for all (Usubsetneq V, Une emptyset ), where (d_A^+(U)) and (d_A^-(U)) denote the number of arcs in A leaving and entering U, respectively. Let ({mathcal {C}}) be a crossing family over ground set V, and let (f:{mathcal {C}}rightarrow {mathbb {Z}}) be a crossing submodular function such that (f(U)ge frac{k}{tau }(d_A^+(U)-d_A^-(U))) for all (Uin {mathcal {C}}). Then D has a k-arc-connected flip J such that (f(U)ge d_J^+(U)-d_J^-(U)) for all (Uin {mathcal {C}}). The result has several applications to Graph Orientations and Combinatorial Optimization. In particular, it strengthens Nash-Williams’ so-called weak orientation theorem, and proves a weaker variant of Woodall’s conjecture on digraphs whose underlying undirected graph is (tau )-edge-connected. The second main result of this paper is even more general. It introduces a sufficient condition for the existence of capacitated integral solutions to the intersection of two submodular flow systems. This sufficient condition implies the classic result of Edmonds and Giles on the box-total dual integrality of a submodular flow system. It also has the consequence that in a weakly connected digraph, the intersection of two submodular flow systems is totally dual integral.

The induced size-Ramsey number (hat{r}_text {ind}^k(H)) of a graph H is the smallest number of edges a (host) graph G can have such that for any k-coloring of its edges, there exists a monochromatic copy of H which is an induced subgraph of G. In 1995, in their seminal paper, Haxell, Kohayakawa and Łuczak showed that for cycles, these numbers are linear for any constant number of colours, i.e., (hat{r}_text {ind}^k(C_n)le Cn) for some (C=C(k)). The constant C comes from the use of the regularity lemma, and has a tower type dependence on k. In this paper we significantly improve these bounds, showing that (hat{r}_text {ind}^k(C_n)le O(k^{102})n) when n is even, thus obtaining only a polynomial dependence of C on k. We also prove (hat{r}_text {ind}^k(C_n)le e^{O(klog k)}n) for odd n, which almost matches the lower bound of (e^{Omega (k)}n). Finally, we show that the ordinary (non-induced) size-Ramsey number satisfies (hat{r}^k(C_n)=e^{O(k)}n) for odd n. This substantially improves the best previous result of (e^{O(k^2)}n), and is best possible, up to the implied constant in the exponent. To achieve our results, we present a new host graph construction which, roughly speaking, reduces our task to finding a cycle of approximate given length in a graph with local sparsity.