Discrete Mathematics最新文献

英文中文

A power sum expansion for the Kromatic symmetric function 罗曼对称函数的幂和展开式

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-12-18 DOI: 10.1016/j.disc.2025.114957

Laura Pierson

<div><div>The chromatic symmetric function <math><msub><mrow><mi>X</mi></mrow><mrow><mi>G</mi></mrow></msub></math> is a symmetric function generalization of the chromatic polynomial of a graph, introduced by Stanley [7]. Stanley [7] gave an expansion formula for <math><msub><mrow><mi>X</mi></mrow><mrow><mi>G</mi></mrow></msub></math> in terms of the power sum symmetric functions <math><msub><mrow><mi>p</mi></mrow><mrow><mi>λ</mi></mrow></msub></math> using the principle of inclusion-exclusion, and Bernardi and Nadeau [1] gave an alternate p-expansion for <math><msub><mrow><mi>X</mi></mrow><mrow><mi>G</mi></mrow></msub></math> in terms of acyclic orientations. Crew, Pechenik, and Spirkl [3] defined the Kromatic symmetric function <math><msub><mrow><mover><mrow><mi>X</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>G</mi></mrow></msub></math> as a K-theoretic analogue of <math><msub><mrow><mi>X</mi></mrow><mrow><mi>G</mi></mrow></msub></math>, constructed in the same way except that each vertex is assigned a nonempty set of colors such that adjacent vertices have nonoverlapping color sets. They defined a K-analogue <math><msub><mrow><mover><mrow><mi>p</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>λ</mi></mrow></msub></math> of the power sum basis and computed the first few coefficients of the <math><mover><mrow><mi>p</mi></mrow><mo>‾</mo></mover></math>-expansion of <math><msub><mrow><mover><mrow><mi>X</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>G</mi></mrow></msub></math> for some small graphs G. They conjectured that the <math><mover><mrow><mi>p</mi></mrow><mo>‾</mo></mover></math>-expansion always has integer coefficients and asked whether there is an explicit formula for these coefficients. In this note, we give a formula for the <math><mover><mrow><mi>p</mi></mrow><mo>‾</mo></mover></math>-expansion of <math><msub><mrow><mover><mrow><mi>X</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>G</mi></mrow></msub></math>, show two ways to compute the coefficients recursively (along with examples), and prove that the coefficients are indeed always integers. In a more recent paper [6], we use our formula from this note to give a combinatorial description of the <math><mover><mrow><mi>p</mi></mrow><mo>‾</mo></mover></math>-coefficients <math><mo>[</mo><msub><mrow><mover><mrow><mi>p</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>λ</mi></mrow></msub><mo>]</mo><msub><mrow><mover><mrow><mi>X</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>G</mi></mrow></msub></math> and a simple characterization of their signs in the case of unweighted graphs.</

色对称函数XG是由Stanley[7]引入的图的色多项式的对称函数推广。Stanley[7]利用包含-排斥原理给出了XG在幂和对称函数pλ下的展开式，Bernardi和Nadeau[1]给出了XG在无环取向下的另一个p展开式。Crew， Pechenik和Spirkl[3]定义了矩阵对称函数X - G作为XG的k理论模拟，以相同的方式构造，除了每个顶点被分配一个非空的颜色集，以便相邻的顶点具有不重叠的颜色集。他们定义了一个幂和基的k -模拟p - λ，并计算了一些小图形G的X - G的p -展开的前几个系数。他们推测p -展开总是有整数系数，并问这些系数是否有一个显式公式。在这篇文章中，我们给出了X - G的p -展开的一个公式，展示了两种递归计算系数的方法（以及例子），并且证明了系数确实总是整数。在最近的一篇论文[6]中，我们使用本笔记中的公式给出了p -系数[p - λ]X - G的组合描述，并在无加权图的情况下给出了它们的符号的简单表征。

引用次数: 0

An improved bound for equitable proper labellings 公平正确标签的改进界

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-12-18 DOI: 10.1016/j.disc.2025.114956

Julien Bensmail , Clara Marcille

For every graph G with size m and no connected component isomorphic to

K_{2}

, we prove that, for

L = (1, 1, 2, 2, \dots, ⌊ m / 2 ⌋ + 2, ⌊ m / 2 ⌋ + 2)

, we can assign labels of L to the edges of G in an injective way so that no two adjacent vertices of G are incident to the same sum of labels. This implies that every such graph with size m can be labelled in an equitable and proper way with labels from

{1, \dots, ⌊ m / 2 ⌋ + 2}

, which improves on a result proved by Haslegrave, and Szabo Lyngsie and Zhong, implying this can be achieved with labels from

{1, \dots, m}

对于每一个大小为m且无连通分量同构于K2的图G，我们证明，对于L=(1,1,2,2，…，⌊m/2⌋+2，⌊m/2⌋+2)，我们可以将L的标记以内射的方式分配给G的边，使得G的两个相邻顶点不隶属于相同的标记和。这意味着每一个大小为m的图都可以用{1，…，⌊m/2⌋+2}的标签来合理地标记，这改进了Haslegrave、Szabo Lyngsie和Zhong证明的结果，表明这可以用{1，…，m}的标签来实现。

{"title":"An improved bound for equitable proper labellings","authors":"Julien Bensmail , Clara Marcille","doi":"10.1016/j.disc.2025.114956","DOIUrl":"10.1016/j.disc.2025.114956","url":null,"abstract":"<div><div>For every graph G with size m and no connected component isomorphic to <math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math>, we prove that, for <math><mi>L</mi><mo>=</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mo>⌊</mo><mi>m</mi><mo>/</mo><mn>2</mn><mo>⌋</mo><mo>+</mo><mn>2</mn><mo>,</mo><mo>⌊</mo><mi>m</mi><mo>/</mo><mn>2</mn><mo>⌋</mo><mo>+</mo><mn>2</mn><mo>)</mo></math>, we can assign labels of L to the edges of G in an injective way so that no two adjacent vertices of G are incident to the same sum of labels. This implies that every such graph with size m can be labelled in an equitable and proper way with labels from <math><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mo>⌊</mo><mi>m</mi><mo>/</mo><mn>2</mn><mo>⌋</mo><mo>+</mo><mn>2</mn><mo>}</mo></math>, which improves on a result proved by Haslegrave, and Szabo Lyngsie and Zhong, implying this can be achieved with labels from <math><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>m</mi><mo>}</mo></math>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 5","pages":"Article 114956"},"PeriodicalIF":0.7,"publicationDate":"2025-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145798383","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A study on token digraphs 符号有向图的研究

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-12-18 DOI: 10.1016/j.disc.2025.114951

Cristina G. Fernandes , Carla N. Lintzmayer , Juan P. Peña , Giovanne Santos , Ana Trujillo-Negrete , Jose Zamora

For a digraph D of order n and an integer

1 \leq k \leq n - 1

, the k-token digraph of D is the digraph whose vertices are all k-subsets of vertices of D and, given two such k-subsets A and B,

(A, B)

is an arc in the k-token digraph whenever

{a} = A ∖ B

{b} = B ∖ A

, and there is an arc

(a, b)

in D. Token digraphs are a generalization of token graphs. In this paper, we study some properties of token digraphs, including strong and unilateral connectivity, kernels, girth, circumference, and Eulerianity. We also extend some known results on the clique and chromatic numbers of k-token graphs, addressing the bidirected clique number and dichromatic number of k-token digraphs. Additionally, we prove that determining whether 2-token digraphs have a kernel is NP-complete.

对于n阶有向图D和整数1≤k≤n−1，D的k- Token有向图是顶点都是D顶点的k个子集的有向图，并且给定两个这样的k-子集a和B，（a， B）是k- Token有向图中的一个弧，当{a}= a∈B, {B}=B∈a，且D中存在一个弧（a， B）时，Token有向图是Token图的推广。本文研究了令牌有向图的一些性质，包括强连通性和单侧连通性、核、周长、周长和欧拉性。我们还推广了关于k-令牌图的团数和色数的一些已知结果，讨论了k-令牌有向图的双向团数和二色数。另外，我们证明了判定2-令牌有向图是否有核是np完全的。

{"title":"A study on token digraphs","authors":"Cristina G. Fernandes , Carla N. Lintzmayer , Juan P. Peña , Giovanne Santos , Ana Trujillo-Negrete , Jose Zamora","doi":"10.1016/j.disc.2025.114951","DOIUrl":"10.1016/j.disc.2025.114951","url":null,"abstract":"<div><div>For a digraph D of order n and an integer <math><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn></math>, the k-token digraph of D is the digraph whose vertices are all k-subsets of vertices of D and, given two such k-subsets A and B, <math><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo></math> is an arc in the k-token digraph whenever <math><mo>{</mo><mi>a</mi><mo>}</mo><mo>=</mo><mi>A</mi><mo>∖</mo><mi>B</mi></math>, <math><mo>{</mo><mi>b</mi><mo>}</mo><mo>=</mo><mi>B</mi><mo>∖</mo><mi>A</mi></math>, and there is an arc <math><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></math> in D. Token digraphs are a generalization of token graphs. In this paper, we study some properties of token digraphs, including strong and unilateral connectivity, kernels, girth, circumference, and Eulerianity. We also extend some known results on the clique and chromatic numbers of k-token graphs, addressing the bidirected clique number and dichromatic number of k-token digraphs. Additionally, we prove that determining whether 2-token digraphs have a kernel is NP-complete.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 5","pages":"Article 114951"},"PeriodicalIF":0.7,"publicationDate":"2025-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145798256","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Some new Bollobás-type inequalities 一些新的Bollobás-type不等式

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-12-17 DOI: 10.1016/j.disc.2025.114948

Erfei Yue

<div><div>A family of disjoint pairs of finite sets <math><mi>P</mi><mo>=</mo><mo>{</mo><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo><mo>|</mo><mi>i</mi><mo>∈</mo><mo>[</mo><mi>m</mi><mo>]</mo><mo>}</mo></math> is called a Bollobás system if <math><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∩</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>≠</mo><mo>∅</mo></math> for every <math><mi>i</mi><mo>≠</mo><mi>j</mi></math>, and a skew Bollobás system if <math><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∩</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>≠</mo><mo>∅</mo></math> for every <math><mi>i</mi><mo><</mo><mi>j</mi></math>. Bollobás proved that for a Bollobás system, the inequality<math><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></munderover><msup><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><mo>+</mo><mo>|</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo></mrow></mtd></mtr><mtr><mtd><mrow><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>⩽</mo><mn>1</mn></math> holds. Hegedüs and Frankl proved that for a skew Bollobás system, we have<math><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></munderover><msup><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><mo>+</mo><mo>|</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo></mrow></mtd></mtr><mtr><mtd><mrow><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>⩽</mo><mn>1</mn><mo>+</mo><mi>n</mi><mo>,</mo></math> provided <math><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>⊆</mo><mo>[</mo><mi>n</mi><mo>]</mo></math>. In this paper, we improve this inequality to<math><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></munderover><msup><mrow><mo>(</mo><mo>(</mo><mn>1</mn><mo>+</mo><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><mo>+</mo><mo>|</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><mo>)</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><mo>+</mo><mo>|</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi>

如果Ai∩Bj≠∅对于每一个i≠j，则一组不相交的有限集对P={(Ai,Bi)|i∈[m]}称为Bollobás系统，如果Ai∩Bj≠∅对于每一个i<；j，则称为一个倾斜Bollobás系统。Bollobás证明了对于一个Bollobás系统，不等式∑i=1m(|Ai|+|Bi||Ai|)−1≤1成立。heged s和Frankl证明了对于一个歪斜Bollobás系统，在给定Ai、Bi的条件下，有∑i=1m(|Ai|+|Bi||Ai|)−1≤1+n。本文利用概率方法将该不等式改进为∑i=1m((1+|Ai|+|Bi|)(|Ai|+|Bi||Ai|))−1≤1。我们还将这一结果推广到对称和偏态情况下的集合划分。

引用次数: 0

The vertex-face chromatic number of almost all nonorientable surfaces 几乎所有不可定向曲面的顶点面色数

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-12-17 DOI: 10.1016/j.disc.2025.114950

Vladimir P. Korzhik

The vertex-face chromatic number

χ_{v f} (N_{q})

of a nonorientable surface

N_{q}

of genus q is the minimum integer m such that the vertices and faces of any map on the surface can be colored by m colors in such a way that adjacent or incident elements receive distinct colors. The known upper bound

Φ (N_{q})

χ_{v f} (N_{q})

differs from Ringel's upper bound on the 1-chromatic number of nonorientable surfaces.

The results are as follows:

(i)
There is a constant $A > 0$ such that for any $q \geq A$ , $Φ (N_{q}) - 1 \leq χ_{v f} (N_{q}) \leq Φ (N_{q})$ .
$(i i)$
Let x be an integer and let $P (x)$ be the number of values of q in the interval $[1, x]$ such that $χ_{v f} (N_{q}) = Φ (N_{q})$ . Then $\lim_{x \to \infty} P (x) / x = 1$ .

不可定向曲面Nq的顶点面色数χvf（Nq）是最小整数m，使得曲面上任何地图的顶点和面都可以用m种颜色着色，从而使相邻或相关元素获得不同的颜色。已知的χvf（Nq）上界Φ（Nq）不同于不可定向曲面1色数上的Ringel上界。结果如下：(i)存在一个常数A>；0，使得对于任意q≥a， Φ(Nq)−1≤χvf(Nq)≤Φ（Nq）。（ii）设x为整数，设P(x)为区间[1，x]中使χvf(Nq)=Φ（Nq）的q值的个数。然后limx→∞⁡P (x) / x = 1。

引用次数: 0

On asymptotically tight bound for the conflict-free chromatic index of nearly regular graphs 近正则图无冲突色指标的渐近紧界

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-12-15 DOI: 10.1016/j.disc.2025.114945

Mateusz Kamyczura, Jakub Przybyło

Let G be a graph of maximum degree Δ which does not contain isolated vertices. An edge coloring c of G is called conflict-free if each edge's closed neighborhood includes a uniquely colored element. The least number of colors admitting such c is called the conflict-free chromatic index of G and denoted

χ_{CF}^{'} (G)

. It is known that in general

χ_{CF}^{'} (G) ⩽ 3 ⌈ \log_{2} Δ ⌉ + 1

, while there is a family of graphs, e.g. the complete graphs, for which

χ_{CF}^{'} (G) ⩾ (1 - o (1)) \log_{2} Δ

. In the present paper we provide the asymptotically tight upper bound

χ_{CF}^{'} (G) ⩽ \log_{2} Δ + O (\log_{2} \log_{2} Δ) = (1 + o (1)) \log_{2} Δ

for regular and nearly regular graphs, which in particular implies that the same bound holds a.a.s. for a random graph

G = G (n, p)

whenever

p ≫ n^{- ε}

for any fixed constant

ε \in (0, 1)

. Our proof is probabilistic and exploits classic results of Hall and Berge. This was inspired by our approach utilized in the particular case of complete graphs, for which we give a more specific upper bound.

设G是一个最大度的图Δ，它不包含孤立的顶点。如果每条边的封闭邻域包含一个唯一的着色元素，则称为无冲突边。允许这种c的颜色的最少数量称为G的无冲突色指数，并表示为χCF ' (G)。众所周知，一般情况下，χCF ' (G)≤3≤≤log2²Δ²+1，而有一类图，例如完整图，其中χCF ' (G)大于或等于（1−0 (1)）log2²Δ。本文给出了正则图和近正则图的渐近紧上界χCF′(G)≤log2 (Δ+O(log2)log2 （Δ）=(1+ O(1))log2 （Δ），特别表明对于任意固定常数ε∈(0,1)，当p≠n−ε时，对于随机图G=G(n,p)，同样的上界也成立。我们的证明是概率性的，并且利用了Hall和Berge的经典结果。这是受到我们在完全图的特殊情况下所使用的方法的启发，我们给出了一个更具体的上界。

引用次数: 0

A generalization of Bender's q-Vandermonde sum 本德q-Vandermonde和的推广

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-12-15 DOI: 10.1016/j.disc.2025.114917

J.G. Bradley-Thrush

A one-parameter generalization is obtained for the non-terminating version of Bender's generalized q-Vandermonde sum. This formula includes the

{}_{6}ϕ_{5}

summation as a special case. Another special case is interpreted combinatorially and given a bijective proof.

得到了Bender广义q-Vandermonde和的非终止型的一个单参数推广。该公式包括作为特殊情况的求和。对另一种特殊情况作了组合解释，并给出了客观证明。

引用次数: 0

Degree sum conditions for optimal 3-restricted arc-connected digraphs 最优3约束圆弧连通有向图的度和条件

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-12-15 DOI: 10.1016/j.disc.2025.114946

Shuaijun Liu, Shangwei Lin, Lin Yang

The k-restricted arc-connectivity of digraphs, as a generalization of the arc-connectivity, is an important index to measure the reliability of directed networks. An arc subset S of a strongly connected digraph D is a k-restricted arc-cut if

D - S

has a strong component

D_{1}

with order at least k such that

D - V (D_{1})

contains a connected subdigraph with order k. The k-restricted arc-connectivity

λ^{k} (D)

of the digraph D is the minimum cardinality over all k-restricted arc-cuts of D. In this paper, we present a degree sum condition for a strongly connected digraph to be optimal in terms of 3-restricted arc-connectivity, and give an example to show that the lower bound on the degree sum in this result is sharp.

有向图的k限制弧连通性是对有向网络可靠性的一种推广，是衡量有向网络可靠性的重要指标。如果D−S具有至少k阶的强分量D1，使得D−V（D1）包含一个k阶的连通子有向图，则强连通有向图D的一个弧子集S是一个k限制弧切。有向图D的k限制弧连通性λk(D)是D的所有k限制弧切上的最小cardinality。本文给出了强连通有向图在3限制弧连通性方面最优的一个度和条件。并举例说明该结果的阶和下界是明显的。

{"title":"Degree sum conditions for optimal 3-restricted arc-connected digraphs","authors":"Shuaijun Liu, Shangwei Lin, Lin Yang","doi":"10.1016/j.disc.2025.114946","DOIUrl":"10.1016/j.disc.2025.114946","url":null,"abstract":"<div><div>The k-restricted arc-connectivity of digraphs, as a generalization of the arc-connectivity, is an important index to measure the reliability of directed networks. An arc subset S of a strongly connected digraph D is a k-restricted arc-cut if <math><mi>D</mi><mo>−</mo><mi>S</mi></math> has a strong component <math><msub><mrow><mi>D</mi></mrow><mrow><mn>1</mn></mrow></msub></math> with order at least k such that <math><mi>D</mi><mo>−</mo><mi>V</mi><mo>(</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math> contains a connected subdigraph with order k. The k-restricted arc-connectivity <math><msup><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>(</mo><mi>D</mi><mo>)</mo></math> of the digraph D is the minimum cardinality over all k-restricted arc-cuts of D. In this paper, we present a degree sum condition for a strongly connected digraph to be optimal in terms of 3-restricted arc-connectivity, and give an example to show that the lower bound on the degree sum in this result is sharp.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 5","pages":"Article 114946"},"PeriodicalIF":0.7,"publicationDate":"2025-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145798328","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Pebbling Cartesian products of C5 C5的卵石笛卡尔积

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-12-15 DOI: 10.1016/j.disc.2025.114943

David S. Herscovici

We imagine a distribution of pebbles on the vertices of a connected graph. Chung defined a pebbling move as the removal of two pebbles from some vertex and the addition of a pebble to an adjacent vertex. Then the pebbling number of a graph G is the smallest number

π (G)

such that every distribution of

π (G)

pebbles allows one pebble to reach any specified, but arbitrary vertex. Graham conjectured that

π (G □ H) \leq π (G) π (H)

for all connected graphs G and H. We show that the pebbling number of

C_{5} □ G

satisfies

π (C_{5} □ G) \leq 5 π (G)

for any connected graph G that satisfies the odd two-pebbling property. In particular,

π (C_{5} □ C_{5} □ C_{5}) = 125

我们想象在连通图的顶点上有一个鹅卵石的分布。Chung将鹅卵石移动定义为从某个顶点移除两个鹅卵石，并在相邻顶点添加一个鹅卵石。那么图G的鹅卵石数是最小的数π(G)，使得π(G)鹅卵石的每个分布都允许一个鹅卵石到达任意指定的顶点。Graham推测对于所有连通图G和H， π(G□H)≤π(G)π(H)。我们证明了对于任何满足奇双铺砾性质的连通图G， C5□G的铺砾数满足π(C5□G)≤5π(G)。其中，π(C5□C5□C5)=125。

引用次数: 0

Periodic points of consecutive-pattern-avoiding stack-sorting maps 避免连续模式的堆栈排序映射的周期点

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-12-15 DOI: 10.1016/j.disc.2025.114947

Ilaria Seidel , Nathan Sun

West's stack-sorting map involves a stack which avoids the permutation 21 consecutively. Defant and Zheng extended this to a consecutive-pattern-avoiding stack-sorting map

S C_{σ}

, where the stack avoids a given permutation σ consecutively. We address one of the main conjectures raised by Defant and Zheng in their dynamical approach to

S C_{σ}

. Specifically, we show that the periodic points of

S C_{σ}

are precisely the permutations that consecutively avoid σ and its reverse.

韦斯特的堆栈排序映射包含一个避免连续排列21的堆栈。Defant和Zheng将其推广到避免连续模式的堆栈排序映射SCσ，其中堆栈连续避免给定排列σ。我们讨论了Defant和Zheng在他们的SCσ动力学方法中提出的一个主要猜想。具体地说，我们证明了SCσ的周期点正是连续避开σ及其逆的排列。

引用次数: 0

首页上一页

下一页尾页

类型

全部化学•材料生命科学医学物理工程技术环境•农林材料科学地球科学法学管理学化学环境科学与生态学计算机科学教育学经济学农林科学人文科学生物学数学物理与天体物理心理学综合性期刊其他工业工程理学历史学农学文学信息工程

数据库

全部 ACS Publications Elsevier ieeexplore Springer The Royal Society of Chemistry Wiley

期刊

Discrete Mathematics

全部 Acc. Chem. Res. ACS Applied Bio Materials ACS Appl. Electron. Mater. ACS Appl. Energy Mater. ACS Appl. Mater. Interfaces ACS Appl. Nano Mater. ACS Appl. Polym. Mater. ACS BIOMATER-SCI ENG ACS Catal. ACS Cent. Sci. ACS Chem. Biol. ACS Chemical Health & Safety ACS Chem. Neurosci. ACS Comb. Sci. ACS Earth Space Chem. ACS Energy Lett. ACS Infect. Dis. ACS Macro Lett. ACS Mater. Lett. ACS Med. Chem. Lett. ACS Nano ACS Omega ACS Photonics ACS Sens. ACS Sustainable Chem. Eng. ACS Synth. Biol. Anal. Chem. BIOCHEMISTRY-US Bioconjugate Chem. BIOMACROMOLECULES Chem. Res. Toxicol. Chem. Rev. Chem. Mater. CRYST GROWTH DES ENERG FUEL Environ. Sci. Technol. Environ. Sci. Technol. Lett. Eur. J. Inorg. Chem. IND ENG CHEM RES Inorg. Chem. J. Agric. Food. Chem. J. Chem. Eng. Data J. Chem. Educ. J. Chem. Inf. Model. J. Chem. Theory Comput. J. Med. Chem. J. Nat. Prod. J PROTEOME RES J. Am. Chem. Soc. LANGMUIR MACROMOLECULES Mol. Pharmaceutics Nano Lett. Org. Lett. ORG PROCESS RES DEV ORGANOMETALLICS J. Org. Chem. J. Phys. Chem. J. Phys. Chem. A J. Phys. Chem. B J. Phys. Chem. C J. Phys. Chem. Lett. Analyst Anal. Methods Biomater. Sci. Catal. Sci. Technol. Chem. Commun. Chem. Soc. Rev. CHEM EDUC RES PRACT CRYSTENGCOMM Dalton Trans. Energy Environ. Sci. ENVIRON SCI-NANO ENVIRON SCI-PROC IMP ENVIRON SCI-WAT RES Faraday Discuss. Food Funct. Green Chem. Inorg. Chem. Front. Integr. Biol. J. Anal. At. Spectrom. J. Mater. Chem. A J. Mater. Chem. B J. Mater. Chem. C Lab Chip Mater. Chem. Front. Mater. Horiz. MEDCHEMCOMM Metallomics Mol. Biosyst. Mol. Syst. Des. Eng. Nanoscale Nanoscale Horiz. Nat. Prod. Rep. New J. Chem. Org. Biomol. Chem. Org. Chem. Front. PHOTOCH PHOTOBIO SCI PCCP Polym. Chem.

﹀