Combinatorica最新文献

A conjecture of Erdős, Graham, Montgomery, Rothschild, Spencer, and Straus states that, with the exception of equilateral triangles, any two-coloring of the plane will have a monochromatic congruent copy of every three-point configuration. This conjecture is known only for special classes of configurations. In this manuscript, we confirm one of the most natural open cases; that is, every two-coloring of the plane admits a monochromatic congruent copy of any 3-term arithmetic progression.

It is well-known that every tournament contains a Hamilton path, and every strongly connected tournament contains a Hamilton cycle. This paper establishes transversal generalizations of these classical results. For a collection (textbf{T}=(T_1,dots ,T_m)) of not-necessarily distinct tournaments on a common vertex set V, an m-edge directed graph (mathcal {D}) with vertices in V is called a (textbf{T})-transversal if there exists a bijection (phi :E(mathcal {D})rightarrow [m]) such that (ein E(T_{phi (e)})) for all (ein E(mathcal {D})). We prove that for sufficiently large m with (m=|V|-1), there exists a (textbf{T})-transversal Hamilton path. Moreover, if (m=|V|) and at least (m-1) of the tournaments (T_1,ldots ,T_m) are assumed to be strongly connected, then there is a (textbf{T})-transversal Hamilton cycle. In our proof, we utilize a novel way of partitioning tournaments which we dub (textbf{H})-partition.

A pure pair of size t in a graph G is a pair A, B of disjoint subsets of V(G), each of cardinality at least t, such that A is either complete or anticomplete to B. It is known that, for every forest H, every graph on (nge 2) vertices that does not contain H or its complement as an induced subgraph has a pure pair of size (Omega (n)); furthermore, this only holds when H or its complement is a forest. In this paper, we look at pure pairs of size (n^{1-c}), where (0<c<1). Let H be a graph: does every graph on (nge 2) vertices that does not contain H or its complement as an induced subgraph have a pure pair of size (Omega (|G|^{1-c}))? The answer is related to the congestion of H, the maximum of (1-(|J|-1)/|E(J)|) over all subgraphs J of H with an edge. (Congestion is nonnegative, and equals zero exactly when H is a forest.) Let d be the smaller of the congestions of H and (overline{H}). We show that the answer to the question above is “yes” if (dle c/(9+15c)), and “no” if (d>c).

For all integers (n ge k > d ge 1), let (m_{d}(k,n)) be the minimum integer (D ge 0) such that every k-uniform n-vertex hypergraph ({mathcal {H}}) with minimum d-degree (delta _{d}({mathcal {H}})) at least D has an optimal matching. For every fixed integer (k ge 3), we show that for (n in k mathbb {N}) and (p = Omega (n^{-k+1} log n)), if ({mathcal {H}}) is an n-vertex k-uniform hypergraph with (delta _{k-1}({mathcal {H}}) ge m_{k-1}(k,n)), then a.a.s. its p-random subhypergraph ({mathcal {H}}_p) contains a perfect matching. Moreover, for every fixed integer (d < k) and (gamma > 0), we show that the same conclusion holds if ({mathcal {H}}) is an n-vertex k-uniform hypergraph with (delta _d({mathcal {H}}) ge m_{d}(k,n) + gamma left( {begin{array}{c}n - d k - dend{array}}right) ). Both of these results strengthen Johansson, Kahn, and Vu’s seminal solution to Shamir’s problem and can be viewed as “robust” versions of hypergraph Dirac-type results. In addition, we also show that in both cases above, ({mathcal {H}}) has at least (exp ((1-1/k)n log n - Theta (n))) many perfect matchings, which is best possible up to an (exp (Theta (n))) factor.

Given a graph G and a parameter r, we define the r-local matroid of G to be the matroid generated by its cycles of length at most r. Extending Whitney’s abstract planar duality theorem from 1932, we prove that for every r the r-local matroid of G is co-graphic if and only if G admits a certain type of embedding in a surface, which we call r-planar embedding. The maximum value of r such that a graph G admits an r-planar embedding is closely related to face-width, and such embeddings for this maximum value of r are quite often embeddings of minimum genus. Unlike minimum genus embeddings, these r-planar embeddings can be computed in polynomial time. This provides the first systematic and polynomially computable method to construct for every graph G a surface so that G embeds in that surface in an optimal way (phrased in our notion of r-planarity).

A storage code on a graph G is a set of assignments of symbols to the vertices such that every vertex can recover its value by looking at its neighbors. We consider the question of constructing large-size storage codes on triangle-free graphs constructed as coset graphs of binary linear codes. Previously it was shown that there are infinite families of binary storage codes on coset graphs with rate converging to 3/4. Here we show that codes on such graphs can attain rate asymptotically approaching 1. Equivalently, this question can be phrased as a version of hat-guessing games on graphs (e.g., Cameron et al., in: Electron J Combin 23(1):48, 2016). In this language, we construct triangle-free graphs with success probability of the players approaching one as the number of vertices tends to infinity. Furthermore, finding linear index codes of rate approaching zero is also an equivalent problem.

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