Combinatorica最新文献

Given a residually connected incidence geometry (Gamma ) that satisfies two conditions, denoted ((B_1)) and ((B_2)), we construct a new geometry (H(Gamma )) with properties similar to those of (Gamma ). This new geometry (H(Gamma )) is inspired by a construction of Lefèvre-Percsy, Percsy and Leemans (Bull Belg Math Soc Simon Stevin 7(4):583–610, 2000). We show how (H(Gamma )) relates to the classical halving operation on polytopes, allowing us to generalize the halving operation to a broader class of geometries, that we call non-degenerate leaf hypertopes. Finally, we apply this generalization to cubic toroids in order to generate new examples of regular hypertopes.

A permutation (pi in mathbb {S}_n) is k-balanced if every permutation of order k occurs in (pi ) equally often, through order-isomorphism. In this paper, we explicitly construct k-balanced permutations for (k le 3), and every n that satisfies the necessary divisibility conditions. In contrast, we prove that for (k ge 4), no such permutations exist. In fact, we show that in the case (k ge 4), every n-element permutation is at least (Omega _n(n^{k-1})) far from being k-balanced. This lower bound is matched for (k=4), by a construction based on the Erdős–Szekeres permutation.

For an odd integer (n = 2d-1), let ({mathcal {B}}_d) be the subgraph of the hypercube (Q_n) induced by the two largest layers. In this paper, we describe the typical structure of proper q-colorings of (V({mathcal {B}}_d)) and give asymptotics on the number of such colorings when q is an even number. The proofs use various tools including information theory (entropy), Sapozhenko’s graph container method and a recently developed method of Jenssen and Perkins that combines Sapozhenko’s graph container lemma with the cluster expansion for polymer models from statistical physics.

A spherical L-code, where (L subseteq [-1,infty )), consists of unit vectors in (mathbb {R}^d) whose pairwise inner products are contained in L. Determining the maximum cardinality (N_L(d)) of an L-code in (mathbb {R}^d) is a fundamental question in discrete geometry and has been extensively investigated for various choices of L. Our understanding in high dimensions is generally quite poor. Equiangular lines, corresponding to (L = {-alpha , alpha }), is a rare and notable solved case. Bukh studied an extension of equiangular lines and showed that (N_L(d) = O_L(d)) for (L = [-1, -beta ] cup {alpha }) with (alpha ,beta > 0) (we call such L-codes “uniacute”), leaving open the question of determining the leading constant factor. Balla, Dräxler, Keevash, and Sudakov proved a “uniform bound” showing (limsup _{drightarrow infty } N_L(d)/d le 2p) for (L = [-1, -beta ] cup {alpha }) and (p = lfloor alpha /beta rfloor + 1). For which ((alpha ,beta )) is this uniform bound tight? We completely answer this question. We develop a framework for studying uniacute codes, including a global structure theorem showing that the Gram matrix has an approximate p-block structure. We also formulate a notion of “modular codes,” which we conjecture to be optimal in high dimensions.

We determine the excluded minors characterising the class of countable graphs that embed into some compact surface.

Abstract polytopes are combinatorial objects that generalise geometric objects such as convex polytopes, maps on surfaces and tilings of the space. Chiral polytopes are those abstract polytopes that admit full combinatorial rotational symmetry but do not admit reflections. In this paper we build chiral polytopes whose facets (maximal faces) are isomorphic to a prescribed regular cubic tessellation of the n-dimensional torus ((n geqslant 2)). As a consequence, we prove that for every (d geqslant 3) there exist infinitely many chiral d-polytopes.

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.