Combinatorica最新文献

In the max–min allocation problem a set P of players are to be allocated disjoint subsets of a set R of indivisible resources, such that the minimum utility among all players is maximized. We study the restricted variant, also known as the Santa Claus problem, where each resource has an intrinsic positive value, and each player covets a subset of the resources. Bezáková and Dani (SIGecom Exch 5(3):11–18, 2005) showed that this problem is NP-hard to approximate within a factor less than 2, consequently a great deal of work has focused on approximate solutions. The principal approach for obtaining approximation algorithms has been via the Configuration LP (CLP) of Bansal and Sviridenko (Proceedings of the 38th ACM Symposium on Theory of Computing, 2006). Accordingly, there has been much interest in bounding the integrality gap of this CLP. The existing algorithms and integrality gap estimations are all based one way or another on the combinatorial augmenting tree argument of Haxell (Graphs Comb 11(3):245–248, 1995) for finding perfect matchings in certain hypergraphs. Our main innovation in this paper is to introduce the use of topological methods, to replace the combinatorial argument of Haxell (Graphs Comb 11(3):245–248, 1995) for the restricted max–min allocation problem. This approach yields substantial improvements in the integrality gap of the CLP. In particular we improve the previously best known bound of 3.808 to 3.534. We also study the ((1,varepsilon ))-restricted version, in which resources can take only two values, and improve the integrality gap in most cases. Our approach applies a criterion of Aharoni and Haxell, and Meshulam, for the existence of independent transversals in graphs, which involves the connectedness of the independence complex. This is complemented by a graph process of Meshulam that decreases the connectedness of the independence complex in a controlled fashion and hence, tailored appropriately to the problem, can verify the criterion. In our applications we aim to establish the flexibility of the approach and hence argue for it to be a potential asset in other optimization problems involving hypergraph matchings.

For two graphs F, H and a positive integer n, the function (f_{F,H}(n)) denotes the largest m such that every H-free graph on n vertices contains an F-free induced subgraph on m vertices. This function has been extensively studied in the last 60 years when F and H are cliques and became known as the Erdős–Rogers function. Recently, Balogh, Chen and Luo, and Mubayi and Verstraëte initiated the systematic study of this function in the case where F is a general graph. Answering, in a strong form, a question of Mubayi and Verstraëte, we prove that for every positive integer r and every (K_{r-1})-free graph F, there exists some (varepsilon _F>0) such that (f_{F,K_r}(n)=O(n^{1/2-varepsilon _F})). This result is tight in two ways. Firstly, it is no longer true if F contains (K_{r-1}) as a subgraph. Secondly, we show that for all (rge 4) and (varepsilon >0), there exists a (K_{r-1})-free graph F for which (f_{F,K_r}(n)=Omega (n^{1/2-varepsilon })). Along the way of proving this, we show in particular that for every graph F with minimum degree t, we have (f_{F,K_4}(n)=Omega (n^{1/2-6/sqrt{t}})). This answers (in a strong form) another question of Mubayi and Verstraëte. Finally, we prove that there exist absolute constants (0<c<C) such that for each (rge 4), if F is a bipartite graph with sufficiently large minimum degree, then (Omega (n^{frac{c}{log r}})le f_{F,K_r}(n)le O(n^{frac{C}{log r}})). This shows that for graphs F with large minimum degree, the behaviour of (f_{F,K_r}(n)) is drastically different from that of the corresponding off-diagonal Ramsey number (f_{K_2,K_r}(n)).

In 2002, D. Fon-Der-Flaass constructed a prolific family of strongly regular graphs. In this paper, we prove that for infinitely many natural numbers n and a positive constant c, this family contains at least (n^{ccdot n^{2/3}}) strongly regular n-vertex graphs X with the same parameters, which satisfy the following condition: an isomorphism between X and any other graph can be verified by the 4-dimensional Weisfeiler-Leman algorithm.

We generalize the (signed) Varchenko matrix of a hyperplane arrangement to complexes of oriented matroids and show that its determinant has a nice factorization. This extends previous results on hyperplane arrangements and oriented matroids.

We prove that in every metric space where no line contains all the points, there are at least (Omega (n^{2/3})) lines. This improves the previous (Omega (sqrt{n})) lower bound on the number of lines in general metric space, and also improves the previous (Omega (n^{4/7})) lower bound on the number of lines in metric spaces generated by connected graphs.

As a first step towards a conjecture of Kahle and Newman, we prove that if (T_n) is a random 2-dimensional determinantal hypertree on n vertices, then

converges to zero in probability. Confirming a conjecture of Linial and Peled, we also prove the analogous statement for the 1-out 2-complex. Our proof relies on the large deviation principle for the Erdős–Rényi random graph by Chatterjee and Varadhan.

The supersaturation problem for a given graph F asks for the minimum number (h_F(n,q)) of copies of F in an n-vertex graph with (textrm{ex}(n,F)+q) edges. Subsequent works by Rademacher, Erdős, and Lovász and Simonovits determine the optimal range of q (which is linear in n) for cliques F such that (h_F(n,q)) equals the minimum number (t_F(n,q)) of copies of F obtained from a maximum F-free n-vertex graph by adding q new edges. A breakthrough result of Mubayi extends this line of research from cliques to color-critical graphs F, and this was further strengthened by Pikhurko and Yilma who established the equality (h_F(n,q)=t_F(n,q)) for (1le qle epsilon _F n) and sufficiently large n. In this paper, we present several results on the supersaturation problem that extend beyond the existing framework. Firstly, we explicitly construct infinitely many graphs F with restricted properties for which (h_F(n,q)<qcdot t_F(n,1)) holds when (ngg qge 4), thus refuting a conjecture of Mubayi. Secondly, we extend the result of Pikhurko–Yilma by showing the equality (h_F(n,q)=t_F(n,q)) in the range (1le qle epsilon _F n) for any member F in a diverse and abundant graph family (which includes color-critical graphs, disjoint unions of cliques (K_r), and the Petersen graph). Lastly, we prove the existence of a graph F for any positive integer s such that (h_F(n,q)=t_F(n,q)) holds when (1le qle epsilon _F n^{1-1/s}), and (h_F(n,q)<t_F(n,q)) when (n^{1-1/s}/epsilon _Fle qle epsilon _F n), indicating that (q=Theta (n^{1-1/s})) serves as the threshold for the equality (h_F(n,q)=t_F(n,q)). We also discuss some additional remarks and related open problems.

Nešetřil and Ossona de Mendez recently proposed a new definition of graph convergence called structural convergence. The structural convergence framework is based on the probability of satisfaction of logical formulas from a fixed fragment of first-order formulas. The flexibility of choosing the fragment allows to unify the classical notions of convergence for sparse and dense graphs. Since the field is relatively young, the range of examples of convergent sequences is limited and only a few methods of construction are known. Our aim is to extend the variety of constructions by considering the gadget construction. We show that, when restricting to the set of sentences, the application of gadget construction on elementarily convergent sequences yields an elementarily convergent sequence. On the other hand, we show counterexamples witnessing that a generalization to the full first-order convergence is not possible without additional assumptions. We give several different sufficient conditions to ensure the full convergence. One of them states that the resulting sequence is first-order convergent if the replaced edges are dense in the original sequence of structures.

An r-regular graph is an r-graph, if every odd set of vertices is connected to its complement by at least r edges. Let G and H be r-graphs. An H-coloring of G is a mapping (f:E(G) rightarrow E(H)) such that each r adjacent edges of G are mapped to r adjacent edges of H. For every (rge 3), let (mathcal H_r) be an inclusion-wise minimal set of connected r-graphs, such that for every connected r-graph G there is an (H in mathcal H_r) which colors G. The Petersen Coloring Conjecture states that (mathcal H_3) consists of the Petersen graph P. We show that if true, then this is a very exclusive situation. Our main result is that either (mathcal H_3 = {P}) or (mathcal H_3) is an infinite set and if (r ge 4), then (mathcal H_r) is an infinite set. In particular, for all (r ge 3), (mathcal H_r) is unique. We first characterize (mathcal H_r) and then prove that if (mathcal H_r) contains more than one element, then it is an infinite set. To obtain our main result we show that (mathcal H_r) contains the smallest r-graphs of class 2 and the smallest poorly matchable r-graphs, and we determine the smallest r-graphs of class 2.

We prove that, for any (B subset {mathbb {R}}), the Cartesian product set (B times B) determines (Omega (|B|^{2+c})) distinct angles.