Combinatorica最新文献

We prove that every properly edge-colored n-vertex graph with average degree at least (32(log 5n)^2) contains a rainbow cycle, improving upon the ((log n)^{2+o(1)}) bound due to Tomon. We also prove that every properly edge-colored n-vertex graph with at least (10^5 k^3 n^{1+1/k}) edges contains a rainbow 2k-cycle, which improves the previous bound (2^{ck^2}n^{1+1/k}) obtained by Janzer. Our method using homomorphism inequalities and a lopsided regularization lemma also provides a simple way to prove the Erdős–Simonovits supersaturation theorem for even cycles, which may be of independent interest.

Let (textbf{G}:=(G_1, G_2, G_3)) be a triple of graphs on the same vertex set V of size n. A rainbow triangle in (textbf{G}) is a triple of edges ((e_1, e_2, e_3)) with (e_iin G_i) for each i and ({e_1, e_2, e_3}) forming a triangle in V. The triples (textbf{G}) not containing rainbow triangles, also known as Gallai colouring templates, are a widely studied class of objects in extremal combinatorics. In the present work, we fully determine the set of edge densities ((alpha _1, alpha _2, alpha _3)) such that if (vert E(G_i)vert > alpha _i n^2) for each i and n is sufficiently large, then (textbf{G}) must contain a rainbow triangle. This resolves a problem raised by Aharoni, DeVos, de la Maza, Montejanos and Šámal, generalises several previous results on extremal Gallai colouring templates, and proves a recent conjecture of Frankl, Győri, He, Lv, Salia, Tompkins, Varga and Zhu.

The amount of hyperplanes that are needed in order to cover the Boolean cube has been studied in various contexts. Linial and Radhakrishnan introduced the notion of essential covers. An essential cover is a collection of hyperplanes that form a minimal cover of the vertices of the hypercube, and every coordinate is influential in at least one of the hyperplanes. Linial and Radhakrishnan proved using algebraic tools that every essential cover of the n-cube must be of size at least (Omega (sqrt{n})). We devise a stronger lower bound method, and show that the size of every essential cover is at least (Omega (n^{0.52})). This result has implications in proof complexity, because essential covers have been used to prove lower bounds for several proof systems.

We prove that every additive set A with energy (E(A)ge |A|^3/K) has a subset (A'subseteq A) of size (|A'|ge (1-varepsilon )K^{-1/2}|A|) such that (|A'-A'|le O_varepsilon (K^{4}|A'|)). This is, essentially, the largest structured set one can get in the Balog–Szemerédi–Gowers theorem.

The chromatic number of a directed graph is the minimum number of induced acyclic subdigraphs that cover its vertex set, and accordingly, the chromatic number of a tournament is the minimum number of transitive subtournaments that cover its vertex set. The neighborhood of an arc uv in a tournament T is the set of vertices that form a directed triangle with arc uv. We show that if the neighborhood of every arc in a tournament has bounded chromatic number, then the whole tournament has bounded chromatic number. This holds more generally for oriented graphs with bounded independence number, and we extend our proof from tournaments to this class of dense digraphs. As an application, we prove the equivalence of a conjecture of El-Zahar and Erdős and a recent conjecture of Nguyen, Scott and Seymour relating the structure of graphs and tournaments with high chromatic number.

We show that (limsup |E(G)|/|V(G)| = 2.5) over all 4-critical planar graphs G, answering a question of Grünbaum from 1988.

Denote by ({mathcal {C}}^-_{ell }) the 3-uniform hypergraph obtained by removing one hyperedge from the tight cycle on (ell ) vertices. It is conjectured that the Turán density of ({mathcal {C}}^-_{5}) is 1/4. In this paper, we make progress toward this conjecture by proving that the Turán density of ({mathcal {C}}^-_{ell }) is 1/4, for every sufficiently large (ell ) not divisible by 3. One of the main ingredients of our proof is a forbidden-subhypergraph characterization of the hypergraphs, for which there exists a tournament on the same vertex set such that every hyperedge is a cyclic triangle in this tournament. A byproduct of our method is a human-checkable proof for the upper bound on the maximum number of almost similar triangles in a planar point set, which was recently proved using the method of flag algebras by Balogh, Clemen, and Lidický.

In this paper we study the fundamental problem of finding small dense subgraphs in a given graph. For a real number (s>2), we prove that every graph on n vertices with average degree (dge s) contains a subgraph of average degree at least s on at most (nd^{-frac{s}{s-2}}(log d)^{O_s(1)}) vertices. This is optimal up to the polylogarithmic factor, and resolves a conjecture of Feige and Wagner. In addition, we show that every graph with n vertices and average degree at least (n^{1-frac{2}{s}+varepsilon }) contains a subgraph of average degree at least s on (O_{varepsilon ,s}(1)) vertices, which is also optimal up to the constant hidden in the O(.) notation, and resolves a conjecture of Verstraëte.

Dirac proved that each n-vertex 2-connected graph with minimum degree at least k contains a cycle of length at least (min {2k, n}). We consider a hypergraph version of this result. A Berge cycle in a hypergraph is an alternating sequence of distinct vertices and edges (v_1,e_2,v_2, ldots , e_c, v_1) such that ({v_i,v_{i+1}} subseteq e_i) for all i (with indices taken modulo c). We prove that for (n ge k ge r+2 ge 5), every 2-connected r-uniform n-vertex hypergraph with minimum degree at least ({k-1 atopwithdelims ()r-1} + 1) has a Berge cycle of length at least (min {2k, n}). The bound is exact for all (kge r+2ge 5).

A d-dimensional framework is a pair (G, p), where (G=(V,E)) is a graph and p is a map from V to ({mathbb {R}}^d). The length of an edge (xyin E) in (G, p) is the distance between p(x) and p(y). A vertex pair ({u,v}) of G is said to be globally linked in (G, p) if the distance between p(u) and p(v) is equal to the distance between q(u) and q(v) for every d-dimensional framework (G, q) in which the corresponding edge lengths are the same as in (G, p). We call (G, p) globally rigid in ({mathbb {R}}^d) when each vertex pair of G is globally linked in (G, p). A pair ({u,v}) of vertices of G is said to be weakly globally linked in G in ({mathbb {R}}^d) if there exists a generic framework (G, p) in which ({u,v}) is globally linked. In this paper we first give a sufficient condition for the weak global linkedness of a vertex pair of a ((d+1))-connected graph G in ({mathbb {R}}^d) and then show that for (d=2) it is also necessary. We use this result to obtain a complete characterization of weakly globally linked pairs in graphs in ({mathbb {R}}^2), which gives rise to an algorithm for testing weak global linkedness in the plane in (O(|V|^2)) time. Our methods lead to a new short proof for the characterization of globally rigid graphs in ({mathbb {R}}^2), and further results on weakly globally linked pairs and globally rigid graphs in the plane and in higher dimensions.