European Journal of Combinatorics最新文献

In this paper, we establish a connection between Apollonian packings and knot theory. We introduce new representations of links realized in the tangency graph of the regular crystallographic sphere packings. Particularly, we prove that any algebraic link can be realized in the cubic section of the orthoplicial Apollonian packing. We use these representations to improve the upper bound on the ball number of an infinite family of alternating algebraic links. Furthermore, the later allow us to reinterpret the correspondence of rational tangles and rational numbers and to reveal geometrically primitive solutions for the Diophantine equation $x^4+y^4+z^4=2t^2$.

We are interested in the distribution of the number of faces across all the $2-$cell embeddings of a graph, which is equivalent to the distribution of genus by Euler’s formula. In order to study this distribution, we consider the local distribution of faces at a single vertex. We show an asymptotic uniformity on this local face distribution which holds for any graph with large vertex degrees.

We use this to study the usual face distribution of the complete graph. We show that in this case, the local face distribution determines the face distribution for almost all of the whole graph. We use this result to show that a portion of the complete graph of size $(1-o\left(1\right))\left|K_n\right|$ has the same face distribution as the set of all permutations, up to parity. Along the way, we prove new character bounds and an asymptotic uniformity on conjugacy class products.

Enumeration of walks with small steps in the quadrant has been a topic of great interest in combinatorics over the last few years. In this article, it is shown how to compute exact asymptotics of the number of such walks with fixed start- and endpoints for orbit-summable models with finite group, up to arbitrary precision. The resulting representation greatly resembles one conjectured for walks starting from the origin in 2020 by Chapon, Fusy and Raschel, differing only in terms appearing due to the periodicity of the model. We will see that the dependency on start- and endpoint is given by discrete polyharmonic functions, which are solutions of ${△}^{n}v=0$ for a discretization $△$ of a Laplace–Beltrami operator. They can be decomposed into a sum of products of lower order polyharmonic functions of either the start- or the endpoint only, which leads to a partial extension of a theorem by Denisov and Wachtel.

For a set $A$ of positive integers, let $P\left(A\right)$ denote the set of all finite subset sums of $A$. In this paper, we completely solve a problem of Chen and Wu by proving that if $B=\{b_1<b_2<\cdots \}$ is a sequence of integers with $b_1\ge 11$, $3b_1+5\le b_2\le 4b_1$, $3b_2+2\le b_3\le 3b_2+b_1$ and $3b_n-b_{n-2}\le b_{n+1}\le 3b_n(n\ge 3)$, then there exists a set of positive integers $A$ for which $P\left(A\right)=N\setminus B$. We also partially answer a problem of Wu by determining the structure of $B=\{b_1<b_2<\cdots \}$ with $b_1>10$ and $b_2>3b_1+4$, for which there exists a set of positive integers $A$ such that