Discrete Mathematics最新文献

英文中文

Remoteness of graphs with given size and connectivity constraints

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-02-26 DOI: 10.1016/j.disc.2025.114451

Peter Dankelmann , Sonwabile Mafunda , Sufiyan Mallu

Let G be a finite, simple connected graph. The average distance of a vertex v of G is the arithmetic mean of the distances from v to all other vertices of G. The remoteness

ρ (G)

of G is the maximum of the average distances of the vertices of G.

In this paper, we give sharp upper bounds on the remoteness of a graph of given order, connectivity and size. We also obtain corresponding bound s for 2-edge-connected and 3-edge-connected graphs, and bounds in terms of order and size for triangle-free graphs.

引用次数: 0

Poset positional games

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-02-25 DOI: 10.1016/j.disc.2025.114455

Guillaume Bagan , Eric Duchêne , Florian Galliot , Valentin Gledel , Mirjana Mikalački , Nacim Oijid , Aline Parreau , Miloš Stojaković

We propose a generalization of positional games, supplementing them with a restriction on the order in which the elements of the board are allowed to be claimed. We introduce poset positional games, which are positional games with an additional structure – a poset on the elements of the board. Throughout the game play, based on this poset and the set of the board elements that are claimed up to that point, we reduce the set of available moves for the player whose turn it is – an element of the board can only be claimed if all the smaller elements in the poset are already claimed.

We proceed to analyze these games in more detail, with a prime focus on the most studied convention, the Maker-Breaker games. First we build a general framework around poset positional games. Then, we perform a comprehensive study of the complexity of determining the game outcome, conditioned on the structure of the family of winning sets on the one side and the structure of the poset on the other.

引用次数: 0

The signless Laplacian spectral radius of book-free graphs

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-02-21 DOI: 10.1016/j.disc.2025.114448

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

Given a graph H, a graph is called H-free if it does not contain H as a subgraph. For a positive integer k, a book

B_{k + 1}

is a graph consisting of

k + 1

triangles sharing a common edge. In this paper, let G be a

B_{k + 1}

-free graph of order

n \geq 49 k^{2} - 22 k + 4

. Then the signless Laplacian spectral radius

q (G) \leq \frac{n + 2 k + \sqrt{{(n - 2 k)}^{2} + 8 k^{2}}}{2}

with equality if and only if

G = {\overline{K}}_{k} \lor H

, where k or

n - k

is even, and H is a

K_{3}

-free k-regular graph of order

n - k

. Furthermore, if k and

n - k

are both odd, the extremal graphs with the maximum signless Laplacian spectral radius are also characterized.

引用次数: 0

Hoffmann-Ostenhof's 3-Decomposition Conjecture

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-02-20 DOI: 10.1016/j.disc.2025.114454

Genghua Fan, Chuixiang Zhou

The 3-Decomposition Conjecture states that every connected cubic graph can be decomposed into a spanning tree, a set of cycles, and a matching. It has been proved independently by different groups of people that every connected cubic graph can be decomposed into a spanning tree, a set of cycles, and a set of vertex-disjoint paths of at most two edges. In this paper, we establish a bound on the number of paths of two edges, proving that every connected cubic graph on n vertices can be decomposed into a spanning tree, a set of cycles, and a set of vertex-disjoint paths of at most two edges such that the number of paths of two edges is at most

\frac{n - 4}{6}

. Our proof is based on a structural analysis, which might provide a new approach to attack the 3-Decomposition Conjecture.

引用次数: 0

The hereditary closure of the unigraphs

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-02-19 DOI: 10.1016/j.disc.2025.114444

Michael D. Barrus , Ann N. Trenk , Rebecca Whitman

A graph with degree sequence π is a unigraph if it is isomorphic to every graph that has degree sequence π. The class of unigraphs is not hereditary and in this paper we study the related hereditary class HCU, the hereditary closure of unigraphs, consisting of all graphs induced in a unigraph. We characterize the class HCU in multiple ways making use of the tools of a decomposition due to Tyshkevich. We also provide a new characterization of the class that consists of unigraphs for which all induced subgraphs are also unigraphs.

引用次数: 0

Some online Maker-Breaker games

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-02-19 DOI: 10.1016/j.disc.2025.114446

Patrick Bennett , Alan Frieze

We consider some Maker-Breaker games of the following flavor. We have some set V of items for purchase. Maker's goal is to purchase some member of a given family

H

of subsets of V as cheaply as possible and Breaker's goal is to make the purchase as expensive as possible. Each player has a pointer and during a player's turn their pointer moves through the items in the order of the permutation until the player decides to take one. We mostly focus on the case where the permutation is random and unknown to the players (it is revealed by the players as their pointers move).

引用次数: 0

Minimum degree k and k-connectedness usually arrive together

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-02-19 DOI: 10.1016/j.disc.2025.114453

Sahar Diskin , Anna Geisler

Let

d, n \in N

be such that

d = ω (1)

, and

d \leq n^{1 - a}

for some constant

a > 0

. Consider a d-regular graph

G = (V, E)

and the random graph process that starts with the empty graph

G (0)

and at each step

G (i)

is obtained from

G (i - 1)

by adding uniformly at random a new edge from E. We show that if G satisfies some (very) mild global edge-expansion, and an almost optimal edge-expansion of sets up to order

O (d \log n)

, then for any constant

k \in N

in the random graph process on G, typically the hitting times of minimum degree at least k and of k-connectedness are equal. This, in particular, covers both d-regular high dimensional product graphs and pseudo-random graphs, and confirms a conjecture of Joos from 2015. We further demonstrate that this result is tight in the sense that there are d-regular n-vertex graphs with optimal edge-expansion of sets up to order

Ω (d)

, for which the probability threshold of minimum degree at least one is different than the probability threshold of connectivity.

{"title":"Minimum degree k and k-connectedness usually arrive together","authors":"Sahar Diskin , Anna Geisler","doi":"10.1016/j.disc.2025.114453","DOIUrl":"10.1016/j.disc.2025.114453","url":null,"abstract":"<div><div>Let <math><mi>d</mi><mo>,</mo><mi>n</mi><mo>∈</mo><mi>N</mi></math> be such that <math><mi>d</mi><mo>=</mo><mi>ω</mi><mo>(</mo><mn>1</mn><mo>)</mo></math>, and <math><mi>d</mi><mo>≤</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>1</mn><mo>−</mo><mi>a</mi></mrow></msup></math> for some constant <math><mi>a</mi><mo>></mo><mn>0</mn></math>. Consider a d-regular graph <math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></math> and the random graph process that starts with the empty graph <math><mi>G</mi><mo>(</mo><mn>0</mn><mo>)</mo></math> and at each step <math><mi>G</mi><mo>(</mo><mi>i</mi><mo>)</mo></math> is obtained from <math><mi>G</mi><mo>(</mo><mi>i</mi><mo>−</mo><mn>1</mn><mo>)</mo></math> by adding uniformly at random a new edge from E. We show that if G satisfies some (very) mild global edge-expansion, and an almost optimal edge-expansion of sets up to order <math><mi>O</mi><mo>(</mo><mi>d</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math>, then for any constant <math><mi>k</mi><mo>∈</mo><mi>N</mi></math> in the random graph process on G, typically the hitting times of minimum degree at least k and of k-connectedness are equal. This, in particular, covers both d-regular high dimensional product graphs and pseudo-random graphs, and confirms a conjecture of Joos from 2015. We further demonstrate that this result is tight in the sense that there are d-regular n-vertex graphs with optimal edge-expansion of sets up to order <math><mi>Ω</mi><mo>(</mo><mi>d</mi><mo>)</mo></math>, for which the probability threshold of minimum degree at least one is different than the probability threshold of connectivity.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 6","pages":"Article 114453"},"PeriodicalIF":0.7,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143445239","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Csikvári's poset and Tutte polynomial

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-02-19 DOI: 10.1016/j.disc.2025.114450

Changxin Ding

Csikvári constructed a poset on trees to prove that several graph functions attain extreme values at the star and the path among the trees on a fixed number of vertices. Reiner and Smith proved that the Tutte polynomials

T (1, y)

of cones over trees, which are the graphs obtained by attaching a cone vertex to a tree, have the described extreme behavior. They further conjectured that the result can be strengthened in terms of Csikvári's poset. We solve this conjecture affirmatively.

引用次数: 0

The Pairing-Hamiltonian property in graph prisms

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-02-19 DOI: 10.1016/j.disc.2025.114441

Marién Abreu , Giuseppe Mazzuoccolo , Federico Romaniello , Jean Paul Zerafa

Let G be a graph of even order, and consider

K_{G}

as the complete graph on the same vertex set as G. A perfect matching of

K_{G}

is called a pairing of G. If for every pairing M of G it is possible to find a perfect matching N of G such that

M \cup N

is a Hamiltonian cycle of

K_{G}

, then G is said to have the Pairing-Hamiltonian property, or PH-property, for short. In 2007, Fink (2007) [4] proved that for every

d \geq 2

, the d-dimensional hypercube

Q_{d}

has the PH-property, thus proving a conjecture posed by Kreweras in 1996. In this paper we extend Fink's result by proving that given a graph G having the PH-property, the prism graph

P (G) = G □ K_{2}

of G has the PH-property as well. Moreover, if G is a connected graph, we show that there exists a positive integer

k_{0}

such that the

k^{th}

-prism of a graph

P^{k} (G)

has the PH-property for all

k \geq k_{0}

{"title":"The Pairing-Hamiltonian property in graph prisms","authors":"Marién Abreu , Giuseppe Mazzuoccolo , Federico Romaniello , Jean Paul Zerafa","doi":"10.1016/j.disc.2025.114441","DOIUrl":"10.1016/j.disc.2025.114441","url":null,"abstract":"<div><div>Let G be a graph of even order, and consider <math><msub><mrow><mi>K</mi></mrow><mrow><mi>G</mi></mrow></msub></math> as the complete graph on the same vertex set as G. A perfect matching of <math><msub><mrow><mi>K</mi></mrow><mrow><mi>G</mi></mrow></msub></math> is called a pairing of G. If for every pairing M of G it is possible to find a perfect matching N of G such that <math><mi>M</mi><mo>∪</mo><mi>N</mi></math> is a Hamiltonian cycle of <math><msub><mrow><mi>K</mi></mrow><mrow><mi>G</mi></mrow></msub></math>, then G is said to have the Pairing-Hamiltonian property, or PH-property, for short. In 2007, Fink (2007) [4] proved that for every <math><mi>d</mi><mo>≥</mo><mn>2</mn></math>, the d-dimensional hypercube <math><msub><mrow><mi>Q</mi></mrow><mrow><mi>d</mi></mrow></msub></math> has the PH-property, thus proving a conjecture posed by Kreweras in 1996. In this paper we extend Fink's result by proving that given a graph G having the PH-property, the prism graph <math><mi>P</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>G</mi><mo>□</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math> of G has the PH-property as well. Moreover, if G is a connected graph, we show that there exists a positive integer <math><msub><mrow><mi>k</mi></mrow><mrow><mn>0</mn></mrow></msub></math> such that the <math><msup><mrow><mi>k</mi></mrow><mrow><mtext>th</mtext></mrow></msup></math>-prism of a graph <math><msup><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></math> has the PH-property for all <math><mi>k</mi><mo>≥</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>0</mn></mrow></msub></math>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 6","pages":"Article 114441"},"PeriodicalIF":0.7,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143437938","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A note on the 2-factor Hamiltonicity Conjecture

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-02-19 DOI: 10.1016/j.disc.2025.114442

Maximilian Gorsky , Theresa Johanni , Sebastian Wiederrecht

The 2-factor Hamiltonicity Conjecture by Funk, Jackson, Labbate, and Sheehan [JCTB, 2003] asserts that all cubic, bipartite graphs in which all 2-factors are Hamiltonian cycles can be built using a simple operation starting from

K_{3, 3}

and the Heawood graph.

We discuss the link between this conjecture and matching theory, in particular by showing that this conjecture is equivalent to the statement that the two exceptional graphs in the conjecture are the only cubic braces in which all 2-factors are Hamiltonian cycles, where braces are connected, bipartite graphs in which every matching of size at most two is contained in a perfect matching. In the context of matching theory this conjecture is especially noteworthy as

K_{3, 3}

and the Heawood graph are both strongly tied to the important class of Pfaffian graphs, with

K_{3, 3}

being the canonical non-Pfaffian graph and the Heawood graph being one of the most noteworthy Pfaffian graphs.

Our main contribution is a proof that the Heawood graph is the only Pfaffian, cubic brace in which all 2-factors are Hamiltonian cycles. This is shown by establishing that, aside from the Heawood graph, all Pfaffian braces contain a cycle of length four, which may be of independent interest.

{"title":"A note on the 2-factor Hamiltonicity Conjecture","authors":"Maximilian Gorsky , Theresa Johanni , Sebastian Wiederrecht","doi":"10.1016/j.disc.2025.114442","DOIUrl":"10.1016/j.disc.2025.114442","url":null,"abstract":"<div><div>The 2-factor Hamiltonicity Conjecture by Funk, Jackson, Labbate, and Sheehan [JCTB, 2003] asserts that all cubic, bipartite graphs in which all 2-factors are Hamiltonian cycles can be built using a simple operation starting from <math><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>3</mn></mrow></msub></math> and the Heawood graph.</div><div>We discuss the link between this conjecture and matching theory, in particular by showing that this conjecture is equivalent to the statement that the two exceptional graphs in the conjecture are the only cubic braces in which all 2-factors are Hamiltonian cycles, where braces are connected, bipartite graphs in which every matching of size at most two is contained in a perfect matching. In the context of matching theory this conjecture is especially noteworthy as <math><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>3</mn></mrow></msub></math> and the Heawood graph are both strongly tied to the important class of Pfaffian graphs, with <math><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>3</mn></mrow></msub></math> being the canonical non-Pfaffian graph and the Heawood graph being one of the most noteworthy Pfaffian graphs.</div><div>Our main contribution is a proof that the Heawood graph is the only Pfaffian, cubic brace in which all 2-factors are Hamiltonian cycles. This is shown by establishing that, aside from the Heawood graph, all Pfaffian braces contain a cycle of length four, which may be of independent interest.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 6","pages":"Article 114442"},"PeriodicalIF":0.7,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143437954","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

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.

﹀