重混合欧拉数：超越连通情形

Eulerian 2-复形

Eulerian 2-Complexes

本文将图的Euler定理推广至2-复形，引入circlet与Eulerian cover概念，并证明强连通纯2-复形中边与面相交偶数次、可分解为circlet并集及存在Eulerian cover三者等价。

关于 GF(p) 上可表示的分裂拟阵的圈

On the circuits of splitting matroids representable over GF(p)

本文将二元拟阵的分裂操作扩展至 GF(p) 上可表示的 p-拟阵，刻画了所得拟阵的圈、基与独立集，并获得了产生欧拉 p-拟阵的充分条件及保持连通性的连通 p-拟阵类。

欧拉数的一个推广

A Generalization of the Eulerian Numbers

本文通过一个自治一阶微分方程推广了各阶欧拉数，并利用其解给出包括伯努利数在内的一些数的积分表示。

Eulerian polynomials and excedance statistics

A formula of Stembridge states that the permutation peak polynomials and descent polynomials are connected via a quadratique transformation. The aim of this paper is to establish the cycle analogue of Stembridge's formula by using cycle peaks and excedances of permutations. We prove a series of new general formulae expressing polynomials counting permutations by various excedance statistics in terms of refined Eulerian polynomials. Our formulae are comparable with Zhuang's generalizations [Adv. in Appl. Math. 90 (2017) 86-144] using descent statistics of permutations. Our methods include permu…

On the algorithmic complexity of finding hamiltonian cycles in special classes of planar cubic graphs

It is a well-known fact that hamiltonicity in planar cubic graphs is an NP-complete problem. This implies that the existence of an A-trail in plane eulerian graphs is also an NP-complete problem even if restricted to planar 3-connected eulerian graphs. In this paper we deal with hamiltonicity in planar cubic graphs G having a facial 2-factor Q via (quasi) spanning trees of faces in G/Q and study the algorithmic complexity of finding such (quasi) spanning trees of faces. We show, in particular, that if Barnette's Conjecture is false, then hamiltonicity in 3-connected planar cubic bipartite grap…

计算并界定某些$4$-正则图类的欧拉定向数量

Computing and Bounding the Number of Eulerian Orientations for Certain Classes of $4$-Regular Graphs

本文改进了$4$-正则图欧拉定向数量的上界，对双连通多重图给出紧界$2^n+2$，对简单图给出指数级上界，并提出分治算法精确计算可分离图，无需穷举。

平面（有向）图划分为一个连通无环集与一个独立集的划分问题

Partitions of planar (oriented) graphs into a connected acyclic and an independent set

围绕欧拉有向平面图的顶点划分交叉问题，研究CAI-划分的存在性，证明两类图存在该划分，并构造反例将已有结果推广至有向场景。

Eulerian properties of hypergraphs

In this paper we study three substructures in hypergraphs that generalize the notion of an Euler tour in a graph. A flag-traversing tour of a hypergraph corresponds to an Euler tour of its incidence graph, hence complete characterization of hypergraphs with an Euler tour follows from Euler's Theorem. An Euler tour is a closed walk that traverses each edge of the hypergraph exactly once; and an Euler family is a family of closed walks that cannot be concatenated and that jointly traverse each edge of the hypergraph exactly once. Lonc and Naroski have shown that the problem of existence of an Eu…

Enumerating alternating matrix spaces over finite fields with explicit coordinates

We initiate the study of enumerating linear subspaces of alternating matrices over finite fields with explicit coordinates. We postulate that this study can be viewed as a linear algebraic analogue of the classical topic of enumerating labelled graphs. To support this viewpoint, we present q-analogues of Gilbert's formula for enumerating connected graphs (Can. J. Math., 1956), and Read's formula for enumerating c-colored graphs (Can. J. Math., 1960). We also develop an analogue of Riddell's formula relating the exponential generating function of graphs with that of connected graphs (Riddell's …

欧拉地图的通用渐近性与精确计数

Universal Asymptotics and Exact Enumeration of Eulerian Maps

研究具有任意度序列的连通、带标记、亏格为 \(g\) 的欧拉地图数量渐近性，结合正交多项式递推系数分析与多变量解析组合学理论，得出通用渐近性及亏格为 \(1\) 时的精确公式，是 \(g\geq 1\) 非正则地图计数问题的首个结果。

关于允许无限筹码发射游戏的最小筹码数

On the minimum number of chips allowing an infinite Chip-firing game

本文针对强连通无环有向多重图，给出了计算导致无限筹码发射游戏所需最小初始筹码数的三个公式，推广了欧拉情形下的已知结果。

Beyond Degree Choosability

Let $G$ be a connected graph with maximum degree $Δ$. Brooks' theorem states that $G$ has a $Δ$-coloring unless $G$ is a complete graph or an odd cycle. A graph $G$ is \emph{degree-choosable} if $G$ can be properly colored from its lists whenever each vertex $v$ gets a list of $d(v)$ colors. In the context of list coloring, Brooks' theorem can be strengthened to the following. Every connected graph $G$ is degree-choosable unless each block of $G$ is a complete graph or an odd cycle; such a graph $G$ is a \emph{Gallai tree}. This degree-choosability result was further strengthened to Alon--Tars…

路径、终点与无限图的分离问题

Paths, Ends and The Separation Problem for Infinite Graphs

本文研究了无限图的分离问题，证明了在有限终点条件下该问题可判定，并展示了其在欧拉路径问题中的应用。

长码中最小码字的最大个数

The maximum number of minimal codewords in long codes

二进制码中最小码字的最大数目的上界遵循拟阵理论。随机编码提供了下限。在本文中，我们比较这些边界与类似的边界的循环码的图。这个问题（在图形的情况下）被认为是在1981年由恩特林格和斯莱特谁问，如果一个连通图的$p$顶点和$q$边可以只略多于$2^{q-p}$周期。本说明中的边界对所有图都是肯定的，除了可能有一些少于$2p+3\log_2(3p)$边的图。我们还得出结论，一个欧拉（偶）图最多有$2^{q-p}$圈，除非该图是一个4-正则图的一个细分，该图是两个Hamilton圈的边不交并，在这种情况下，它可能有多达$2^{q-p}+p$圈。