Combinatorica最新文献

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.

Given integers (n> k > 0), and a set of integers (L subset [0, k-1]), an L-system is a family of sets (mathcal {F}subset left( {begin{array}{c}[n] kend{array}}right) ) such that (|F cap F'| in L) for distinct (F, F'in mathcal {F}). L-systems correspond to independent sets in a certain generalized Johnson graph G(n, k, L), so that the maximum size of an L-system is equivalent to finding the independence number of the graph G(n, k, L). The Lovász number (vartheta (G)) is a semidefinite programming approximation of the independence number (alpha ) of a graph G. In this paper, we determine the leading order term of (vartheta (G(n, k, L))) of any generalized Johnson graph with k and L fixed and (nrightarrow infty ). As an application of this theorem, we give an explicit construction of a graph G on n vertices with a large gap between the Lovász number and the Shannon capacity c(G). Specifically, we prove that for any (epsilon > 0), for infinitely many n there is a generalized Johnson graph G on n vertices which has ratio (vartheta (G)/c(G) = Omega (n^{1-epsilon })), which improves on all known constructions. The graph G a fortiori also has ratio (vartheta (G)/alpha (G) = Omega (n^{1-epsilon })), which greatly improves on the best known explicit construction.

Let G be a bridgeless cubic graph. In 2023, the three authors solved a conjecture (also known as the (S_4)-Conjecture) made by Mazzuoccolo in 2013: there exist two perfect matchings of G such that the complement of their union is a bipartite subgraph of G. They actually show that given any (1^+)-factor F (a spanning subgraph of G such that its vertices have degree at least 1) and an arbitrary edge e of G, there exists a perfect matching M of G containing e such that (Gsetminus (Fcup M)) is bipartite. This is a step closer to comprehend better the Fan–Raspaud Conjecture and eventually the Berge–Fulkerson Conjecture. The (S_4)-Conjecture, now a theorem, is also the weakest assertion in a series of three conjectures made by Mazzuoccolo in 2013, with the next stronger statement being: there exist two perfect matchings of G such that the complement of their union is an acyclic subgraph of G. Unfortunately, this conjecture is not true: Jin, Steffen, and Mazzuoccolo later showed that there exists a counterexample admitting 2-cuts. Here we show that, despite of this, every cyclically 3-edge-connected cubic graph satisfies this second conjecture.

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.