Discrete Mathematics最新文献

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Weak saturation numbers for the union of disjoint graphs 不相交图并集的弱饱和数

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-05-01 Epub Date: 2026-01-27 DOI: 10.1016/j.disc.2026.115011

Yu Zhang, Rong-Xia Hao, Zhen He, Jianbing Liu

<div><div>Let F and H be two graphs. A spanning subgraph G of F is said to be weakly <math><mo>(</mo><mi>F</mi><mo>,</mo><mi>H</mi><mo>)</mo></math>-saturated if there exists an ordering <math><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>t</mi></mrow></msub></math> of the edges in <math><mi>E</mi><mo>(</mo><mi>F</mi><mo>)</mo><mo>∖</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math> such that, for each <math><mi>i</mi><mo>∈</mo><mo>[</mo><mi>t</mi><mo>]</mo></math>, the addition of <math><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi></mrow></msub></math> to <math><mi>G</mi><mo>+</mo><mo>{</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>}</mo></math> creates a new copy of H containing <math><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi></mrow></msub></math>. The weak saturation number of H with respect to F is defined as <math><mi>w</mi><mi>s</mi><mi>a</mi><mi>t</mi><mo>(</mo><mi>F</mi><mo>,</mo><mi>H</mi><mo>)</mo><mo>=</mo><mi>min</mi><mo>⁡</mo><mo>{</mo><mo>|</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>:</mo><mi>G</mi><mtext> is weakly </mtext><mo>(</mo><mi>F</mi><mo>,</mo><mi>H</mi><mo>)</mo><mtext>-saturated</mtext><mo>}</mo></math>. Kronenberg et al. (2021) [7] determined the exact values of <math><mi>w</mi><mi>s</mi><mi>a</mi><mi>t</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>)</mo></math> and <math><mi>w</mi><mi>s</mi><mi>a</mi><mi>t</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo></math>. In this paper, we generalize previous results by determining the exact values of <math><mi>w</mi><mi>s</mi><mi>a</mi><mi>t</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>)</mo></math> and <math><mi>w</mi><mi>s</mi><mi>a</mi><mi>t</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo></math> for <math><mi>r</mi><mo>≥</mo><mn>1</mn></math>, and provide both upper and lower bounds for <math><mi>w</mi><mi>s</mi><mi>a</mi><mi>t</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>)</mo></math>. Additionally, we determine <math><mi>w</mi><mi>s</mi><mi>a</mi><mi>t</mi><mo>(</mo><mi>n</mi><mo>,</mo><munderover><mo>⋃</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mr

设F和H是两个图。F的生成子图G是弱（F，H）饱和的，如果存在E(F)∈E(G)中的边的排序e1，…，et，使得对于每一个i∈[t]， ei加上G+{e1，…，ei−1}产生一个包含ei的H的新副本。H相对于F的弱饱和数定义为wsat(F，H)=min (|E(G)|:G是弱（F，H）饱和}。Kronenberg et al.(2021)[7]确定了wsat（n,Kt,t）和wsat（n,Kt,t+1）的确切值。本文通过确定r≥1时wsat（n,rKt,t）和wsat（n,rKt,t+1）的精确值，推广了前人的结果，并给出了wsat（n,rKs,t）的上界和下界。此外，我们确定了不相交完全图并集的wsat(n,∈i=1qKti)，这改进了Faudree等人关于wsat（n,qKt）的已知结果。

引用次数: 0

On Ward numbers and increasing Schröder trees 关于病房数和增加Schröder树

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-05-01 Epub Date: 2025-12-30 DOI: 10.1016/j.disc.2025.114965

Elena L. Wang , Guoce Xin

The Ward numbers

W (n, k)

combinatorially enumerate set partitions with block sizes ≥2 and phylogenetic trees (total partition trees). We prove that

W (n, k)

also counts increasing Schröder trees by verifying they satisfy Ward's recurrence. We construct a direct type-preserving bijection between total partition trees and increasing Schröder trees, complementing known type-preserving bijections to set partitions (including Chen's decomposition for increasing Schröder trees). Weighted generalizations extend these bijections to enriched increasing Schröder trees and Schröder trees, yielding new links to labeled rooted trees. Finally, we deduce a functional equation for weighted increasing Schröder trees, whose solution using Chen's decomposition leads to a combinatorial interpretation of a Lagrange inversion variant.

Ward数W（n,k）组合枚举块大小≥2的集合分区和系统发育树（总分区树）。我们通过验证W（n,k）满足Ward递归来证明W（n,k）也算增加Schröder树。我们在总分区树和增加的Schröder树之间构造了一个直接保型双射，补充了已知的保型双射来设置分区（包括Chen的增加Schröder树的分解）。加权泛化将这些双向映射扩展到丰富的增长Schröder树和Schröder树，生成到标记有根树的新链接。最后，我们推导了加权增加Schröder树的函数方程，其解使用Chen的分解导致拉格朗日反转变体的组合解释。

引用次数: 0

The burning game on graphs 图表上的燃烧游戏

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-05-01 Epub Date: 2026-02-02 DOI: 10.1016/j.disc.2026.115027

Nina Chiarelli , Vesna Iršič Chenoweth , Marko Jakovac , William B. Kinnersley , Mirjana Mikalački

Motivated by the burning and cooling processes, the burning game is introduced. Two players (Burner and Staller) play the game on a graph G by alternately selecting vertices of G to burn; as in the burning process, burning vertices spread fire to unburned neighbors. Burner aims to burn all vertices of G as quickly as possible, while Staller wants the process to last as long as possible. If both players play optimally, then the number of time steps needed to burn the whole graph G is the game burning number

b_{g} (G)

if Burner makes the first move, and the Staller-start game burning number

b_{g}^{'} (G)

if Staller starts.

In this paper, basic bounds on

b_{g} (G)

are given and several fundamental properties of the burning game established. Graphs with small game burning numbers are characterized and the game is studied on paths and cycles. An analogue of the burning number conjecture for the burning game is also considered. Finally, it is shown that the problem of determining whether or not

b_{g} (G) \leq k

is NP-hard.

在燃烧和冷却过程的启发下，介绍了燃烧游戏。两名玩家（Burner和Staller）在图形G上轮流选择要燃烧的顶点来玩游戏；在燃烧过程中，燃烧的顶点将火传播到未燃烧的邻居。Burner的目标是尽可能快地燃烧G的所有顶点，而Staller希望这个过程持续尽可能长的时间。如果两个玩家都处于最佳状态，那么燃烧整个图形G所需的时间步数就是游戏燃烧数bg(G)（如果Burner先采取行动），而Staller开始游戏燃烧数bg(G)（如果Staller开始）。本文给出了bg(G)的基本界，并建立了燃烧对策的几个基本性质。对游戏燃烧数较小的图进行了表征，并在路径和循环上对游戏进行了研究。我们还考虑了燃烧游戏的燃烧数猜想的类比。最后，证明了确定bg(G)≤k是否为np困难问题。

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引用次数: 0

Spectral extrema of graphs: Forbidden cliques and star forests 图的谱极值：禁团和星林

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-04-01 Epub Date: 2025-12-23 DOI: 10.1016/j.disc.2025.114954

Ming-Zhu Chen , Ya-Lei Jin , Peng-Li Zhang

Let

F

be a family of graphs. A graph is called

F

-free if it does not contain any member of

F

as a subgraph. A star forest

\cup_{i = 1}^{s + 1} S_{d_{i}}

is a forest whose all components are stars, where

S_{d_{i}}

is a star of order

d_{i} + 1

. In this paper, for

k \geq 2

s \geq 1

and

n \geq 100 {(\sum_{i = 1}^{s + 1} d_{i} + 3 s + 1)}^{6}

, we obtain the maximum spectral radius of

{K_{k + 1}, \cup_{i = 1}^{s + 1} S_{d_{i}}}

-free graphs of order n, where

d_{1} \geq \dots \geq d_{s + 1} \geq 1

. Moreover, we also characterize the extremal graphs.

设F是一个图族。如果一个图不包含F中的任何成员作为子图，则称为F自由图。星林∪i=1s+1Sdi是一个所有成分都是星的星林，其中Sdi是di+1阶的星。在本文中，对于k≥2，s≥1,n≥100(∑i=1s+1di+3s+1)6，我们得到了n阶自由图{Kk+1，∪i=1s+1Sdi}的最大谱半径，其中d1≥⋯≥ds+1≥1。此外，我们还对极值图进行了刻画。

引用次数: 0

Secure domination in P5-free graphs P5-free图中的安全支配

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-04-01 Epub Date: 2025-12-02 DOI: 10.1016/j.disc.2025.114905

Uttam K. Gupta , Michael A. Henning , Paras Vinubhai Maniya , Dinabandhu Pradhan

A dominating set of a graph G is a set

S \subseteq V (G)

such that every vertex in

V (G) ∖ S

has a neighbor in S, where two vertices are neighbors if they are adjacent. A secure dominating set of G is a dominating set S of G with the additional property that for every vertex

v \in V (G) ∖ S

, there exists a neighbor u of v in S such that

(S ∖ {u}) \cup {v}

is a dominating set of G. The secure domination number of G, denoted by

γ_{s} (G)

, is the minimum cardinality of a secure dominating set of G. We prove that if G is a

P_{5}

-free graph, then

γ_{s} (G) \leq \frac{3}{2} α (G)

, where

α (G)

denotes the independence number of G. We further show that if G is a connected

(P_{5}, H)

-free graph for some

H \in {P_{3} \cup P_{1}, K_{2} \cup 2 K_{1}, paw, C_{4}}

, then

γ_{s} (G) \leq \max {3, α (G)}

. We also show that if G is a

(P_{3} \cup P_{2})

-free graph, then

γ_{s} (G) \leq α (G) + 1

图G的支配集是一个集S⊥V(G)，满足V(G)∑S中的每个顶点在S中有一个邻居，其中两个顶点相邻为邻居。G的安全控制集是G的控制集S，其附加性质是对于每个顶点v∈v (G)∑S，在S中存在一个v的邻居u，使得(S∈{u})∪{v}是G的控制集。G的安全控制数，记作γs(G)，是G的安全控制集的最小cardinality。我们证明如果G是P5-free图，则γs(G)≤32α(G)，其中α(G)表示G的独立数，进一步证明了对于某些H∈{P3∪P1，K2∪2K1,paw,C4}，如果G是连通（P5，H）自由图，则γs(G)≤max (3,α(G)}。我们还证明了如果G是一个（P3∪P2）自由图，那么γs(G)≤α(G)+1。

引用次数: 0

On representation functions related to the partition 论与分区相关的表示函数

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-04-01 Epub Date: 2025-12-02 DOI: 10.1016/j.disc.2025.114908

Fang-Gang Xue , Xue-Qin Cao

<div><div>Let <math><mi>Z</mi></math> be the set of integers and <math><msub><mrow><mi>N</mi></mrow><mrow><mn>0</mn></mrow></msub></math> (resp. <math><mi>N</mi></math>) the set of non-negative (resp. positive) integers. For a nonempty set of integers A and integers n, <math><mi>h</mi><mo>≥</mo><mn>2</mn></math>, denote <math><msub><mrow><mi>r</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>h</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math> by the number of representations of n of the form <math><mi>n</mi><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub></math>, where <math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≤</mo><mo>⋯</mo><mo>≤</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub></math> and <math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi>A</mi></math> for <math><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>h</mi></math>, and <math><msub><mrow><mi>d</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math> by the number of <math><mo>(</mo><mi>a</mi><mo>,</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></math> with <math><mi>a</mi><mo>,</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi>A</mi></math> such that <math><mi>n</mi><mo>=</mo><mi>a</mi><mo>−</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup></math>. Following Lev's work, we prove that there is a partition <math><mi>Z</mi><mo>=</mo><msubsup><mrow><mo>⋃</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></msubsup><msub><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow></msub></math> of the set of all integers such that, for each <math><msub><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow></msub></math>, we have <math><msub><mrow><mi>r</mi></mrow><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><mn>2</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mn>1</mn></math> for all integers n and <math><msub><mrow><mi>d</mi></mrow><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mn>1</mn></math> for all positive integers n. As a main result, we also prove that, for any integers <math><mi>h</mi><mo>≥</mo><mn>2</mn></math> and <math><mi>m</mi><mo>≥</mo><mn>2</mn></math>, there exists a set T of integers with the density <math><mn>1</mn><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>m</mi></mrow></mfrac></mat

设Z为整数的集合，N0 (p。N)非负的(resp。积极的)整数。对于整数a和整数n， h≥2的非空集合，用n的表示形式n=a1+a2+⋯+ah的个数表示rA，h(n)，其中a1≤，≤ah，且对于i=1,2,⋯,h, dA(n)用（a,a’）的个数表示，其中a，a’∈a使得n=a - a’。根据Lev的工作，我们证明了所有整数集合的一个分区Z= k=1∞Ak，使得对于每一个Ak，我们有rAk，2(n)=1，对于所有正整数n，我们有dAk(n)=1。作为一个主要结果，我们还证明了对于任何整数h≥2，m≥2，存在一个密度为1 - 1m的整数集合T，对于任何函数f:Z→N0′{∞}，f−1(0)=T，存在一个整数集合a，满足rA，h(n)=f(n)对于所有n∈Z。这提供了一个与内特森问题相关的特定密度结果。

引用次数: 0

Ramsey numbers of trees versus generalized wheels 树的拉姆齐数与广义轮

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-04-01 Epub Date: 2025-12-09 DOI: 10.1016/j.disc.2025.114938

Yanbo Zhang , Yaojun Chen , Yunqing Zhang

Given two graphs G and H, the Ramsey number

R (G, H)

is the smallest positive integer r such that every graph on r vertices contains G as a subgraph or its complement contains H as a subgraph. Let

T_{n}

denote a tree on n vertices, and let

W_{s, m}

denote a generalized wheel, obtained by joining each vertex of the complete graph

K_{s}

to every vertex of the cycle

C_{m}

. For

s \geq 2

m \geq 2

, and sufficiently large n, Chng, Tan, and Wong (Discrete Math., 2021) conjectured that

R (T_{n}, W_{s, 2 m}) = (s + 1) (n - 1) + 1 .

In this note, we confirm this conjecture in a stronger form by providing a linear lower bound on n in terms of m for which the equality holds.

给定两个图G和H，拉姆齐数R（G，H）是最小的正整数R，使得在R个顶点上的每个图都包含G作为子图或其补包含H作为子图。设Tn表示有n个顶点的树，设Ws，m表示一个广义轮，通过将完全图k的每个顶点与循环Cm的每个顶点连接而得到。对于s≥2，m≥2，和足够大的n， cheng, Tan, and Wong（离散数学）。, 2021)推测thatR (Tn Ws 2米)= (s + 1) (n−1)+ 1。在这篇笔记中，我们以一种更强的形式证实了这个猜想，我们用m给出了n的线性下界，在这个下界中等式成立。

{"title":"Ramsey numbers of trees versus generalized wheels","authors":"Yanbo Zhang , Yaojun Chen , Yunqing Zhang","doi":"10.1016/j.disc.2025.114938","DOIUrl":"10.1016/j.disc.2025.114938","url":null,"abstract":"<div><div>Given two graphs G and H, the Ramsey number <math><mi>R</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></math> is the smallest positive integer r such that every graph on r vertices contains G as a subgraph or its complement contains H as a subgraph. Let <math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub></math> denote a tree on n vertices, and let <math><msub><mrow><mi>W</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>m</mi></mrow></msub></math> denote a generalized wheel, obtained by joining each vertex of the complete graph <math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub></math> to every vertex of the cycle <math><msub><mrow><mi>C</mi></mrow><mrow><mi>m</mi></mrow></msub></math>. For <math><mi>s</mi><mo>≥</mo><mn>2</mn></math>, <math><mi>m</mi><mo>≥</mo><mn>2</mn></math>, and sufficiently large n, Chng, Tan, and Wong (Discrete Math., 2021) conjectured that<math><mi>R</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>W</mi></mrow><mrow><mi>s</mi><mo>,</mo><mn>2</mn><mi>m</mi></mrow></msub><mo>)</mo><mo>=</mo><mo>(</mo><mi>s</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mn>1</mn><mo>.</mo></math> In this note, we confirm this conjecture in a stronger form by providing a linear lower bound on n in terms of m for which the equality holds.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 4","pages":"Article 114938"},"PeriodicalIF":0.7,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145747783","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

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 : 2026-04-01 Epub 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

Extremal graphs for disjoint union of vertex-critical graphs 顶点临界图不相交并的极值图

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-04-01 Epub Date: 2026-01-02 DOI: 10.1016/j.disc.2025.114973

Wenqian Zhang

For a graph F, let

EX (n, F)

be the set of F-free graphs of order n with the maximum number of edges. The graph F is called vertex-critical, if the deletion of its some vertex induces a graph with smaller chromatic number. For example, an odd wheel (obtained by connecting a vertex to a cycle of even length) is a vertex-critical graph with chromatic number 3.

For

h \geq 2

, let

F_{1}, F_{2}, . . ., F_{h}

be vertex-critical graphs with the same chromatic number. Let

\cup_{1 \leq i \leq h} F_{i}

be the disjoint union of them. In this paper, we characterize the graphs in

EX (n, \cup_{1 \leq i \leq h} F_{i})

, when there is a proper order among the graphs

F_{1}, F_{2}, . . ., F_{h}

. This solves a conjecture (on extremal problem for disjoint union of odd wheels) proposed by Xiao and Zamora [16].

对于图F，设EX（n，F）为边数最大的n阶无F图的集合。如果图F的某些顶点被删除，则图F的色数变小，则称为顶点临界图。例如，奇轮（通过将一个顶点连接到一个偶数长度的循环得到）是一个色数为3的顶点临界图。当h≥2时，设F1，F2，…， h是具有相同色数的顶点临界图。设∪1≤i≤hFi是它们的不相交并。在本文中，当图F1，F2，…，Fh之间存在适当的序时，我们刻画了EX（n,∪1≤i≤hFi）中的图。这解决了Xiao和Zamora[16]提出的一个猜想（关于奇轮不接合并的极值问题）。

引用次数: 0

A note on the stability of the potential function 关于势函数稳定性的注解

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-04-01 Epub Date: 2025-11-24 DOI: 10.1016/j.disc.2025.114902

Jian-Hua Yin

Given a graph H, a graphic sequence π is potentially H-graphic if there is a realization of π containing H as a subgraph. Erdős et al. introduced the following problem: determine the minimum even integer

σ (H, n)

such that each n-term graphic sequence with sum at least

σ (H, n)

is potentially H-graphic. The parameter

σ (H, n)

is known as the potential function of H, and can be viewed as a degree sequence variant of the classical extremal function ex

(n, H)

. Ferrara et al. (2016) [3] established an upper bound on

σ (H, n)

and determined

σ (H, n)

asymptotically for an arbitrary graph H. Yin (2020) [6] also obtained an upper bound on

σ (H, n)

. Erbes et al. (2018) [1] defined a stability concept for the potential number, which is a natural analogue to the stability of the classical extremal function given by Simonovits. They gave a sufficient condition for a graph H to be stable with respect to the potential function, and characterized the stability of those graphs H that contain an induced subgraph of order

α (H) + 1

with exactly one edge. In this paper, we further characterize the stability of those graphs H that contain an induced subgraph of order

α (H) + 1

with exactly t independent edges for

1 \leq t \leq ⌊ \frac{α + 1}{2} ⌋

. Therefore, the stability for all graphs H is characterized completely.

给定一个图H，一个图序列π是潜在的H图，如果π包含H作为子图的实现。Erdős等人引入了以下问题：确定最小偶数σ(H,n)，使得每个求和至少为σ(H,n)的n项图序列都是潜在的H图。参数σ(H,n)称为H的势函数，可以看作是经典极值函数ex（n，H）的阶序列变体。Ferrara et al.(2016)[3]建立了σ(H,n)的上界，并渐近地确定了任意图H的σ(H,n)。Yin(2020)[6]也获得了σ(H,n)的上界。Erbes et al.(2018)[1]定义了势数的稳定性概念，这是Simonovits给出的经典极值函数稳定性的自然类比。他们给出了图H相对于势函数稳定的充分条件，并刻画了含有α(H)+1阶的诱导子图H只有一条边的图H的稳定性。在本文中，我们进一步刻画了含有α(H)+1阶的诱导子图H的稳定性，这些图H具有恰好t条独立边，且1≤t≤⌊α+12⌋。因此，所有图H的稳定性被完全刻画。

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