Combinatorica最新文献

In this paper, we prove that the Fourier entropy of an n-dimensional boolean function f can be upper-bounded by (O(I(f)+ sum limits _{kin [n]}I_k(f)log frac{1}{I_k(f)})), where I(f) is its total influence and (I_k(f)) is the influence of the k-th coordinate. There is no strict quantitative relationship between our bound with the known bounds for the Fourier-Min-Entropy-Influence conjecture (O(I(f)log I(f))) and (O(I(f)^2)). The proof is elementary and uses iterative bounds on moments of Fourier coefficients over different levels to estimate the Fourier entropy as its derivative.

A majority edge-coloring of a graph without pendant edges is a coloring of its edges such that, for every vertex v and every color (alpha ), there are at most as many edges incident to v colored with (alpha ) as with all other colors. We extend some known results for finite graphs to infinite graphs, also in the list setting. In particular, we prove that every infinite graph without pendant edges has a majority edge-coloring from lists of size 4. Another interesting result states that every infinite graph without vertices of finite odd degrees admits a majority edge-coloring from lists of size 2. As a consequence of our results, we prove that line graphs of any cardinality admit majority vertex-colorings from lists of size 2, thus confirming the Unfriendly Partition Conjecture for line graphs.

A subset (mathcal {C}subseteq {0,1,2}^n) is said to be a trifferent code (of block length n) if for every three distinct codewords (x,y, z in mathcal {C}), there is a coordinate (iin {1,2,ldots ,n}) where they all differ, that is, ({x(i),y(i),z(i)}) is same as ({0,1,2}). Let T(n) denote the size of the largest trifferent code of block length n. Understanding the asymptotic behavior of T(n) is closely related to determining the zero-error capacity of the (3/2)-channel defined by Elias (IEEE Trans Inform Theory 34(5):1070–1074, 1988), and is a long-standing open problem in the area. Elias had shown that (T(n)le 2times (3/2)^n) and prior to our work the best upper bound was (T(n)le 0.6937 times (3/2)^n) due to Kurz (Example Counterexample 5:100139, 2024). We improve this bound to (T(n)le c times n^{-2/5}times (3/2)^n) where c is an absolute constant.

In this paper we give anticoncentration bounds for sums of independent random vectors in finite-dimensional vector spaces. In particular, we asymptotically establish a conjecture of Leader and Radcliffe (SIAM J Discrete Math 7:90–101, 1994) and a question of Jones (SIAM J Appl Math 34:1–6, 1978). The highlight of this work is an application of the strong perfect graph theorem by Chudnovsky et al. (Ann Math 164:51–229, 2006) in the context of anticoncentration.

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.