Discrete Mathematics最新文献_第8页

On the two-distance embedding in real Euclidean space of coherent configuration of type (2,2;3)

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2024-12-27 DOI: 10.1016/j.disc.2024.114378

Eiichi Bannai , Etsuko Bannai , Chin-Yen Lee , Ziqing Xiang , Wei-Hsuan Yu

{"title":"On the two-distance embedding in real Euclidean space of coherent configuration of type (2,2;3)","authors":"Eiichi Bannai , Etsuko Bannai , Chin-Yen Lee , Ziqing Xiang , Wei-Hsuan Yu","doi":"10.1016/j.disc.2024.114378","DOIUrl":"10.1016/j.disc.2024.114378","url":null,"abstract":"<div><div>Finding the maximum cardinality of a 2-distance set in Euclidean space is a classical problem in geometry. Lisoněk in 1997 constructed a maximum 2-distance set in <math><msup><mrow><mi>R</mi></mrow><mrow><mn>8</mn></mrow></msup></math> with 45 points. That 2-distance set constructed by Lisoněk has a distinguished structure of a coherent configuration of type <math><mo>(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>;</mo><mn>3</mn><mo>)</mo></math> and is embedded in two concentric spheres in <math><msup><mrow><mi>R</mi></mrow><mrow><mn>8</mn></mrow></msup></math>. In this paper we study whether there exists any other similar embedding of a coherent configuration of type <math><mo>(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>;</mo><mn>3</mn><mo>)</mo></math> as a 2-distance set in <math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math>, without assuming any restriction on the size of the set. We prove that there exists no such example other than that of Lisoněk. The key ideas of our proof are as follows: (i) study the geometry of the embedding of the coherent configuration in Euclidean spaces and to derive diophantine equations coming from this embedding. (ii) solve diophantine equations with certain additional conditions of integrality of some parameters of the combinatorial structure by using the method of auxiliary equations.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 4","pages":"Article 114378"},"PeriodicalIF":0.7,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143171245","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 short note on Chen-Qian theorem

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2024-12-27 DOI: 10.1016/j.disc.2024.114371

Jiyou Li, Yanghongbo Zhou

引用次数: 0

Existential closure in uniform hypergraphs

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2024-12-23 DOI: 10.1016/j.disc.2024.114372

Andrea C. Burgess , Robert D. Luther , David A. Pike

引用次数: 0

A note on the random triadic process

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2024-12-20 DOI: 10.1016/j.disc.2024.114374

Fang Tian , Yiting Yang

引用次数: 0

Larger nearly orthogonal sets over finite fields

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2024-12-19 DOI: 10.1016/j.disc.2024.114373

Ishay Haviv , Sam Mattheus , Aleksa Milojević , Yuval Wigderson

{"title":"Larger nearly orthogonal sets over finite fields","authors":"Ishay Haviv , Sam Mattheus , Aleksa Milojević , Yuval Wigderson","doi":"10.1016/j.disc.2024.114373","DOIUrl":"10.1016/j.disc.2024.114373","url":null,"abstract":"<div><div>For a field <math><mi>F</mi></math> and integers d and k, a set <math><mi>A</mi><mo>⊆</mo><msup><mrow><mi>F</mi></mrow><mrow><mi>d</mi></mrow></msup></math> is called k-nearly orthogonal if its members are non-self-orthogonal and every <math><mi>k</mi><mo>+</mo><mn>1</mn></math> vectors of <math><mi>A</mi></math> include an orthogonal pair. We prove that for every prime p there exists some <math><mi>δ</mi><mo>=</mo><mi>δ</mi><mo>(</mo><mi>p</mi><mo>)</mo><mo>></mo><mn>0</mn></math>, such that for every field <math><mi>F</mi></math> of characteristic p and for all integers <math><mi>k</mi><mo>≥</mo><mn>2</mn></math> and <math><mi>d</mi><mo>≥</mo><mi>k</mi></math>, there exists a k-nearly orthogonal set of at least <math><msup><mrow><mi>d</mi></mrow><mrow><mi>δ</mi><mo>⋅</mo><mi>k</mi><mo>/</mo><mi>log</mi><mo>⁡</mo><mi>k</mi></mrow></msup></math> vectors of <math><msup><mrow><mi>F</mi></mrow><mrow><mi>d</mi></mrow></msup></math>. The size of the set is optimal up to the <math><mi>log</mi><mo>⁡</mo><mi>k</mi></math> term in the exponent. We further prove two extensions of this result. In the first, we provide a large set <math><mi>A</mi></math> of non-self-orthogonal vectors of <math><msup><mrow><mi>F</mi></mrow><mrow><mi>d</mi></mrow></msup></math> such that for every two subsets of <math><mi>A</mi></math> of size <math><mi>k</mi><mo>+</mo><mn>1</mn></math> each, some vector of one of the subsets is orthogonal to some vector of the other. In the second extension, every <math><mi>k</mi><mo>+</mo><mn>1</mn></math> vectors of the produced set <math><mi>A</mi></math> include <math><mi>ℓ</mi><mo>+</mo><mn>1</mn></math> pairwise orthogonal vectors for an arbitrary fixed integer <math><mn>1</mn><mo>≤</mo><mi>ℓ</mi><mo>≤</mo><mi>k</mi></math>. The proofs involve probabilistic and spectral arguments and the hypergraph container method.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 4","pages":"Article 114373"},"PeriodicalIF":0.7,"publicationDate":"2024-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143170183","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

The number of perfect matchings in a brick

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2024-12-18 DOI: 10.1016/j.disc.2024.114365

Fuliang Lu, Huali Pan

引用次数: 0

Binary [n,(n ± 1)/2] cyclic codes with good minimum distances from sequences

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2024-12-18 DOI: 10.1016/j.disc.2024.114369

Xianhong Xie , Yaxin Zhao , Zhonghua Sun , Xiaobo Zhou

引用次数: 0

Properly colored C4→'s in arc-colored complete and complete bipartite digraphs

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2024-12-17 DOI: 10.1016/j.disc.2024.114367

Mengyu Duan , Binlong Li , Shenggui Zhang

引用次数: 0

On 1-2-3 Conjecture-like problems in 2-edge-coloured graphs

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2024-12-17 DOI: 10.1016/j.disc.2024.114368

Julien Bensmail , Hervé Hocquard , Clara Marcille , Sven Meyer

{"title":"On 1-2-3 Conjecture-like problems in 2-edge-coloured graphs","authors":"Julien Bensmail , Hervé Hocquard , Clara Marcille , Sven Meyer","doi":"10.1016/j.disc.2024.114368","DOIUrl":"10.1016/j.disc.2024.114368","url":null,"abstract":"<div><div>The well-known 1-2-3 Conjecture asks whether almost all graphs can have their edges labelled with <math><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></math> so that any two adjacent vertices are distinguished w.r.t. the sums of their incident labels. This conjecture has attracted increasing attention over the last years, with many of its aspects of interest being investigated by several authors. In early 2023, Keusch proposed a full solution to the 1-2-3 Conjecture.</div><div>Among other aspects of interest, several works introduced and studied ways of generalising such distinguishing labellings and the 1-2-3 Conjecture to structures more general than graphs, such as digraphs and hypergraphs. In the current work, we introduce two new variants for 2-edge-coloured graphs (having negative and positive edges), in which, through labellings, pairs of adjacent vertices are considered distinguished if and only if the differences between their incident positive and negative sums are different. The difference between the two variants we introduce is that, in one of them, this distinction must be met even when considering the absolute value of these differences.</div><div>We investigate how these two variants connect, and how they relate to the original problem. For each of the two variants, we also establish upper bounds on the minimum number of consecutive labels that suffice to design a distinguishing labelling of almost any 2-edge-coloured graph. This leads us to raise some conjectures on this minimum, which, as support, we prove for some families of 2-edge-coloured graphs. We also investigate weaker versions of these conjectures, where one can choose the polarity of the edges.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 4","pages":"Article 114368"},"PeriodicalIF":0.7,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143171268","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

The intersection density of non-quasiprimitive groups of degree 3p

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2024-12-17 DOI: 10.1016/j.disc.2024.114364

Roghayeh Maleki , Andriaherimanana Sarobidy Razafimahatratra

{"title":"The intersection density of non-quasiprimitive groups of degree 3p","authors":"Roghayeh Maleki , Andriaherimanana Sarobidy Razafimahatratra","doi":"10.1016/j.disc.2024.114364","DOIUrl":"10.1016/j.disc.2024.114364","url":null,"abstract":"<div><div>The intersection density of a finite transitive group <math><mi>G</mi><mo>≤</mo><mi>Sym</mi><mo>(</mo><mi>Ω</mi><mo>)</mo></math> is the rational number <math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math> given by the ratio between the maximum size of a subset of G in which any two permutations agree on some elements of Ω and the order of a point stabilizer of G. In 2022, Meagher asked whether <math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>∈</mo><mo>{</mo><mn>1</mn><mo>,</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mn>3</mn><mo>}</mo></math> for any transitive group G of degree 3p, where <math><mi>p</mi><mo>≥</mo><mn>5</mn></math> is an odd prime. If <math><mi>G</mi><mo>≤</mo><mi>Sym</mi><mo>(</mo><mi>Ω</mi><mo>)</mo></math> is transitive such that <math><mo>|</mo><mi>Ω</mi><mo>|</mo><mo>=</mo><mn>3</mn><mi>p</mi></math>, then it is known that <math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mn>1</mn></math> whenever (a) G is primitive or (b) G is imprimitive and admits a block of size p or at least two G-invariant partitions of Ω. In order to answer Meagher's question, it is left to analyze the intersection density of groups G admitting a unique G-invariant partition <math><mi>B</mi></math> whose blocks are of size 3. If G is such a group and <math><mover><mrow><mi>G</mi></mrow><mo>‾</mo></mover></math> is the group induced by the action of G on <math><mi>B</mi></math>, then we denote the kernel of the canonical epimorphism <math><mi>G</mi><mo>→</mo><mover><mrow><mi>G</mi></mrow><mo>‾</mo></mover></math> by <math><mi>ker</mi><mo>⁡</mo><mo>(</mo><mi>G</mi><mo>→</mo><mover><mrow><mi>G</mi></mrow><mo>‾</mo></mover><mo>)</mo></math>. The subgroup <math><mi>ker</mi><mo>⁡</mo><mo>(</mo><mi>G</mi><mo>→</mo><mover><mrow><mi>G</mi></mrow><mo>‾</mo></mover><mo>)</mo></math> is trivial if and only if G is quasiprimitive.</div><div>It is shown in this paper that the answer to Meagher's question is affirmative for non-quasiprimitive groups of degree 3p, unless possibly when <math><mi>p</mi><mo>=</mo><mi>q</mi><mo>+</mo><mn>1</mn></math> is a Fermat prime and Ω admits a unique G-invariant partition <math><mi>B</mi></math> whose blocks are of size 3 such that the induced action <math><mover><mrow><mi>G</mi></mrow><mo>‾</mo></mover></math> is an almost simple group with socle equal to <math><msub><mrow><mi>PSL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 4","pages":"Article 114364"},"PeriodicalIF":0.7,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143170182","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