实现有向图图化度序列的Havel-Hakimi型简单算法

具有强Havel-Hakimi性质的图

Graphs with the strong Havel-Hakimi property

Havel-Hakimi算法迭代地将图的度序列减少到零的列表。如Favaron、Mahéo和Saclé所示，产生的零的数量，称为剩余，是图的独立数的下界。我们称一个图具有强Havel-Hakimi性质，如果在它的每个诱导子图中，删除任何最大度的顶点都以Havel-Hakimi算法相同的方式减少度序列。我们刻画了具有这一性质的图（包括所有的阈值图和母图）的最小禁止诱导子图。我们进一步表明，这些图的剩余等于独立数，和一个自然的贪婪算法总是产生一个最大的独立集。

有向图度序列的实现算法

Algorithms for realizing degree sequences of directed graphs

构造无向图度序列实现的Havel-Hakimi算法在文献中得到了广泛的应用。Kleitman和Wang的一个结果将Havel-Hakimi算法推广到有向图的度序列。在本文中，我们更进一步，并描述了修改Kleitman和王的算法，这是一个更自然扩张的Havel-Hakimi的算法，在这个意义上，我们的扩展可以等同于Havel-Hakimi的算法时，度序列有相等的程度和甚至度总和。我们确定特殊的程度序列，称为有向3-循环锚定，这是定义不清的算法，并迫使一个特定的局部结构上的所有有向图实现。我们给出了这些实现的结构特征，以及定义不明确的度序列的特征，导致一个定义良好的算法。

公约数图的Havel--Hakimi残差：完整渐近展开与素数计数结构

Havel--Hakimi Residues of Common-Divisor Graphs: Complete Asymptotics and Prime-Counting Structure

本文针对顶点为2到n的公约数图，确定其Havel--Hakimi残差的完整渐近展开，证明无标号图可通过单纯真孪生类确定素数计数函数，所得结果与黎曼假设等价。

On the Domination Order among Elimination Sequences

In 1991, it was shown by Favaron, Mahéo, and Saclé that the residue, which is defined as the number of zeros remaining when the Havel-Hakimi algorithm is applied to a degree sequence, yields a lower bound on the independence number of any graph realising the sequence. In 1996, Triesch simplified and generalised the result by introducing elimination sequences. It was proved in 1973 by Kleitman and Wang that for any graphic sequence all elimination algorithms, i.e. laying-off vertices in any order, preserve that the sequence is graphic and terminate in a sequence of zeros. We now prove that for …

On the Degree Sequences of Multigraphs with Edge Additions and Deletions

The degree sequence of a graph is a numerical method to characterize the properties of graphs. Generalized forms of degree sequences exist for complete graphs and complete graphs. Nikolopolus et al. characterized the number of spanning trees from edge deletions and edge additions. Instead of investigating the number of spanning trees of graphs that arise from edge additions and deletions, we sought to characterize degree sequences of such graphs. We conjecture a characterization for the degree sequence of the addition and edge deletion for many families of graphs including threshold graphs and…

关于无向、有向和双向图的度序列

On degree sequences of undirected, directed, and bidirected graphs

双向图是有向图和无向图的推广，因为它的边在每个节点上都是局部定向的。考虑（局部）方向的节点度的自然概念是净度。在本文中，我们扩展了以下四个主题从（非）有向图双向有向图：-Erdens-Gallai型结果：净度序列的特征，- Havel-Hakimi型结果：度保持操作的完整集合，-极值度序列：唯一可实现序列的特征，和-枚举方面：计数公式的净度序列。为了强调的相似性和差异，他们的（联合国）有向同行，我们简要调查的无向设置，我们给一个彻底的帐户有向图的离散几何的程度序列的重点。特别地，我们确定了有向图的紧且唯一可实现的度序列。