Discrete Mathematics最新文献

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Disconnected forbidden pairs force supereulerian graphs to be hamiltonian 断开的禁止对迫使超立体图成为哈密顿图

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-31 DOI: 10.1016/j.disc.2024.114301

Qiang Wang , Liming Xiong

A graph is said to be supereulerian if it has a spanning eulerian subgraph, i.e., a spanning connected even subgraph. A graph is called hamiltonian if it contains a spanning cycle. A graph is said to be

{R, S}

-free if it does not contain R or S as an induced subgraph. Yang et al. characterized all pairs of connected graphs

R, S

such that every supereulerian

{R, S}

-free graph is hamiltonian. In this paper, we consider disconnected forbidden graph

R, S

. We characterize all pairs of disconnected graphs

R, S

such that every supereulerian

{R, S}

-free graph of sufficiently large order is hamiltonian. Applying this result, we also characterize all forbidden pairs for the existence of a Hamiltonian cycle in 2-edge connected graphs.

如果一个图有一个跨越的优勒子图，即一个跨越的连通偶数子图，则称该图为超优勒图。如果一个图包含一个跨循环，则称为哈密顿图。如果一个图不包含 R 或 S 作为诱导子图，则称其为无{R,S}图。Yang等人描述了所有连通图R,S的特征，即每个无超循环{R,S}图都是哈密顿图。在本文中，我们考虑断开的禁止图 R,S。我们描述了所有成对的断开图 R,S 的特征，即每个阶数足够大的无超线性 {R,S} 图都是哈密顿图。应用这一结果，我们还描述了在 2 边相连图中存在哈密顿循环的所有禁止图对。

引用次数: 0

Rigid frameworks with dilation constraints 具有扩张约束的刚性框架

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-31 DOI: 10.1016/j.disc.2024.114304

Sean Dewar , Anthony Nixon , Andrew Sainsbury

We consider the rigidity and global rigidity of bar-joint frameworks in Euclidean d-space under additional dilation constraints in specified coordinate directions. In this setting we obtain a complete characterisation of generic rigidity. We then consider generic global rigidity. In particular, we provide an algebraic sufficient condition and a weak necessary condition. We also construct a large family of globally rigid frameworks and conjecture a combinatorial characterisation when most coordinate directions have dilation constraints.

我们考虑了欧几里得 d 空间中棒关节框架在指定坐标方向的额外扩张约束下的刚度和全局刚度。在这种情况下，我们得到了一般刚性的完整特征。然后，我们考虑一般全局刚度。特别是，我们提供了一个代数充分条件和一个弱必要条件。我们还构建了一个庞大的全局刚性框架族，并猜想了当大多数坐标方向都有扩张约束时的组合特征。

引用次数: 0

A note on locating-dominating sets in twin-free graphs 关于无孪生图中定位支配集的说明

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-30 DOI: 10.1016/j.disc.2024.114297

Nicolas Bousquet , Quentin Chuet , Victor Falgas–Ravry , Amaury Jacques , Laure Morelle

In this short note, we prove that every twin-free graph on n vertices contains a locating-dominating set of size at most

⌈ \frac{5}{8} n ⌉

. This improves the earlier bound of

⌊ \frac{2}{3} n ⌋

due to Foucaud, Henning, Löwenstein and Sasse from 2016, and makes some progress towards the well-studied locating-dominating conjecture of Garijo, González and Márquez.

在这篇短文中，我们证明了 n 个顶点上的每个无孪生图都包含一个大小至多为⌈58n⌉的定位支配集。这改进了 Foucaud、Henning、Löwenstein 和 Sasse 在 2016 年提出的⌊23n⌋ 的早期约束，并在实现 Garijo、González 和 Márquez 的定位支配猜想方面取得了一些进展。

引用次数: 0

Generation of 3-connected, planar line graphs 生成三连平面线图

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-30 DOI: 10.1016/j.disc.2024.114302

Phoebe Hollowbread-Smith , Riccardo W. Maffucci

We classify and construct all line graphs that are 3-polytopes (planar and 3-connected). Apart from a few special cases, they are all obtained starting from the medial graphs of cubic (i.e., 3-regular) 3-polytopes, by applying two types of graph transformations. This is similar to the generation of other subclasses of 3-polytopes [6], [13].

我们对所有属于 3 多面体（平面和 3 连接）的线图进行了分类和构造。除少数特例外，它们都是从立方（即 3-不规则）3-多面体的中间图开始，通过应用两种图形变换得到的。这与生成其他子类的 3 多面体[6], [13]类似。

引用次数: 0

Light 3-faces in 3-polytopes without adjacent triangles 无相邻三角形的 3 多面体中的光 3 面

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-24 DOI: 10.1016/j.disc.2024.114299

O.V. Borodin , A.O. Ivanova

Over the last decades, a lot of research has been devoted to structural and coloring problems on plane graphs that are sparse in this or that sense.

In this note we deal with the densest among sparse 3-polytopes, namely those having no adjacent 3-cycles. Borodin (1996) proved that such 3-polytopes have a vertex of degree at most 4 and, moreover, an edge with the degree-sum of its end-vertices at most 9, where both bounds are sharp.

d (v)

denote the degree of a vertex v. An edge

e = x y

in a 3-polytope is an

(i, j)

-edge if

d (x) \leq i

and

d (y) \leq j

. The well-known (3,5;4,4)-Archimedean solid corresponds to a plane quadrangulation in which every edge joins a 3-vertex with a 5-vertex.

We prove that every 3-polytope with neither adjacent 3-cycles nor

(3, 5)

-edges has a 3-face with the degree-sum of its incident vertices (weight) at most 16, which bound is sharp.

在过去的几十年里，很多人致力于研究在这种或那种意义上稀疏的平面图的结构和着色问题。在本论文中，我们将讨论稀疏 3 多面体中最密集的一种，即没有相邻 3 循环的多面体。鲍罗丁（Borodin，1996 年）证明了这种 3 多面体的顶点阶数最多为 4，而且边的末端顶点阶数总和最多为 9，这两个界限都很尖锐。我们证明了每一个既没有相邻 3 循环也没有 (3,5) 边的 3 多面体都有一个其入射顶点度数和（权重）最多为 16 的 3 面，这个约束是尖锐的。

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引用次数: 0

Counting spanning trees of multiple complete split-like graph containing a given spanning forest 对包含给定生成林的多个完整分裂样图的生成树进行计数

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-24 DOI: 10.1016/j.disc.2024.114300

Chenlin Yang, Tao Tian

The multiple complete split-like graph

M C S_{b, s}^{a}

is the join of an empty graph

{\overline{K}}_{a}

and s copies of complete graph

K_{b}

. In this article, we obtain the formulas for the number of spanning trees of

M C S_{b, s}^{a}

containing a given spanning forest when

s = 1

and 2. Particularly, when

s = 1

, our result derives the number of spanning trees of complete split graph containing a given spanning forest, thereby extending Moon's result [19].

多重完整分裂样图 MCSb,sa 是空图 K‾a 和完整图 Kb 的 s 个副本的连接。在本文中，我们得到了当 s=1 和 2 时，MCSb,sa 中包含给定生成林的生成树的数量公式。特别是当 s=1 时，我们的结果推导出了包含给定生成林的完整分裂图的生成树数，从而扩展了 Moon 的结果 [19]。

引用次数: 0

Large matchings in maximal 1-planar graphs 最大 1 平面图中的大匹配

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-24 DOI: 10.1016/j.disc.2024.114288

Therese Biedl , John Wittnebel

It is well-known that every maximal planar graph has a matching of size at least

\frac{n + 8}{3}

n \geq 14

. In this paper, we investigate similar matching-bounds for maximal 1-planar graphs, i.e., graphs that can be drawn such that every edge has at most one crossing. In particular we show that every 3-connected simple-maximal 1-planar graph has a matching of size at least

\frac{2 n + 6}{5}

; the bound decreases to

\frac{3 n + 14}{10}

if the graph need not be 3-connected. We also give (weaker) bounds when the graph comes with a fixed 1-planar drawing or is not simple. All our bounds are tight in the sense that some graph that satisfies the restrictions has no bigger matching.

众所周知，如果 n≥14 ，则每个最大平面图的匹配大小至少为 n+83。在本文中，我们将研究最大 1-planar 图的类似匹配边界，即可以画出每条边最多有一个交叉点的图。特别是，我们证明了每个 3 连接的简单最大 1-planar 图都有一个大小至少为 2n+65 的匹配；如果图不需要是 3 连接的，则边界会减小到 3n+1410。当图形带有固定的 1-planar 图或不简单时，我们也给出了（较弱的）边界。我们的所有约束都很严格，因为满足限制条件的某些图没有更大的匹配。

引用次数: 0

The decycling number of a line graph 折线图的去周期数

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-22 DOI: 10.1016/j.disc.2024.114291

Mingyuan Ma, Han Ren

The decycling number of a graph G, denoted by

\nabla (G)

, is the number of vertices in a minimum decycling set of G. The line graph of G is denoted by

L (G)

. In this paper we show that

\nabla (L (G)) = β (G) + μ (G) - 1

, where

β (G)

is the cycle rank of G and

μ (G)

is the path partition number of G. In particular,

\nabla (L (G)) = β (G)

if and only if G has a Hamilton path, and

\nabla (L (G)) \leq \frac{2 n}{3} - \frac{2}{3}

if G is a cubic graph with n vertices, where

n \geq 10

. If

L (G)

is a planar graph, then we prove that

\nabla (L (G)) \leq \frac{| V (L (G)) |}{2}

, which means that the conjecture proposed by Albertson and Berman in 1979 that the decycling number of any planar graph H is at most

\frac{| V (H) |}{2}

holds for a planar line graph. If G is a connected graph of order n which is 2-cell embedded on the orientable surface

\sum_{g}

(or the non-orientable surface

\sum_{k}^{'}

), then we show that

\nabla (L (G)) \leq 2 n + l - 7 + 6 g

(or

2 n + l - 7 + 3 k

) if G has a spanning tree with l leaves. Our bounds are tight for

l = 2

图 G 的去循环数用 ∇(G) 表示，是 G 的最小去循环集合中的顶点数。本文证明了∇(L(G))=β(G)+μ(G)-1，其中β(G)是 G 的循环秩，μ(G)是 G 的路径分割数。尤其是，当且仅当 G 有一条汉密尔顿路径时，∇(L(G))=β(G)；当 G 是有 n 个顶点的立方图时，∇(L(G))≤2n3-23，其中 n≥10 。如果 L(G) 是平面图，那么我们证明∇(L(G))≤|V(L(G))|2，这意味着阿尔伯森和伯曼在 1979 年提出的猜想，即任何平面图 H 的去环数最多为|V(H)|2，对于平面线图是成立的。如果 G 是一个阶数为 n 的连通图，并且在可定向曲面 ∑g （或不可定向曲面 ∑k′）上有 2 个单元嵌入，那么我们证明，如果 G 有一棵有 l 个叶子的生成树，∇(L(G))≤2n+l-7+6g（或 2n+l-7+3k）。当 l=2 时，我们的边界是紧密的。

引用次数: 0

A transient equivalence between Aldous-Broder and Wilson's algorithms and a two-stage framework for generating uniform spanning trees 阿尔多斯-布罗德算法和威尔逊算法之间的瞬时等价关系以及生成均匀生成树的两阶段框架

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-22 DOI: 10.1016/j.disc.2024.114285

Igor Nunes , Giulio Iacobelli , Daniel Ratton Figueiredo

The Aldous-Broder and Wilson are two well-known algorithms for generating uniform spanning trees (USTs) based on random walks. This work studies their transient relationship by introducing the notion of branches—paths generated by the two algorithms on particular stopping times, in order to show that the trees built by the two algorithms when running on a complete graph are statistically equivalent on these stopping times. This leads to a hybrid algorithm that can generate USTs faster than either of the two algorithms. The idea is generalized to a two-stage framework to generate USTs on arbitrary graphs. The feasibility of the framework is shown through various examples, including some edge transitive graphs where the average running time can be 25% smaller than Wilson to generate USTs. Results obtained through numerical simulations of the framework on complete graphs and hypercubes illustrate the findings.

Aldous-Broder 算法和 Wilson 算法是基于随机游走生成均匀生成树（UST）的两种著名算法。这项研究通过引入这两种算法在特定停止时间生成的分支路径概念来研究它们之间的瞬时关系，从而证明这两种算法在完整图上运行时生成的树在这些停止时间上具有统计学等价性。这就产生了一种混合算法，它生成 UST 的速度比两种算法中的任何一种都快。这一想法被推广到一个两阶段框架，用于在任意图上生成 UST。该框架的可行性通过各种实例得到了证明，其中包括一些边缘传递图，在这些图上生成 UST 所需的平均运行时间比 Wilson 算法少 25%。该框架在完整图和超立方体上的数值模拟结果也说明了这些发现。

引用次数: 0

2-Distance (Δ + 1)-coloring of sparse graphs using the potential method 使用势能法对稀疏图进行 2-Distance (Δ + 1) 着色

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-21 DOI: 10.1016/j.disc.2024.114292

Hoang La, Mickael Montassier

A 2-distance k-coloring of a graph is a proper k-coloring of the vertices where vertices at distance at most 2 cannot share the same color. We prove the existence of a 2-distance (

Δ + 1

)-coloring for graphs with maximum average degree less than

\frac{18}{7}

and maximum degree

Δ \geq 7

. As a corollary, every planar graph with girth at least 9 and

Δ \geq 7

admits a 2-distance

(Δ + 1)

-coloring. The proof uses the potential method to reduce new configurations compared to classic approaches on 2-distance coloring.

图的 2-distance k-coloring 是顶点的适当 k-coloring ，其中顶点的距离最多为 2，不能共享相同的颜色。我们证明了最大平均度数小于 187 且最大度数Δ≥7 的图的 2-distance (Δ+1)-coloring 的存在。作为推论，每个周长至少为 9 且 Δ≥7 的平面图都有一个 2-distance (Δ+1)-coloring 。与经典的 2-距离着色方法相比，证明使用了势法来减少新的配置。

引用次数: 0

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