Discrete Mathematics最新文献_第9页

Quasi-orthogonal extension of symmetric matrices 对称矩阵的拟正交扩展

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2025-04-03 DOI: 10.1016/j.disc.2025.114517

Abderrahim Boussaïri , Brahim Chergui , Zaineb Sarir , Mohamed Zouagui

引用次数: 0

On proper hamiltonicity and proper (even) pancyclicity of arc-colored complete (balanced bipartite) digraphs 关于圆弧色完全（平衡二部）有向图的适当哈密性和适当（偶）环性

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2025-04-03 DOI: 10.1016/j.disc.2025.114507

Mengyu Duan , Zhiwei Guo , Binlong Li , Shenggui Zhang

引用次数: 0

The graph of a family of functions over quadratic extensions of finite fields 有限域的二次扩展上的一组函数的图

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2025-04-01 DOI: 10.1016/j.disc.2025.114500

Claude Gravel , Daniel Panario , Hugo Teixeira

{"title":"The graph of a family of functions over quadratic extensions of finite fields","authors":"Claude Gravel , Daniel Panario , Hugo Teixeira","doi":"10.1016/j.disc.2025.114500","DOIUrl":"10.1016/j.disc.2025.114500","url":null,"abstract":"<div><div>Brochero and Teixeira (2023) <span><span>[4]</span></span> showed the behavior of the functional graph of <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>a</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>q</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>+</mo><mi>a</mi><msup><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> over quadratic extensions of finite fields explicitly for <span><math><mi>a</mi><mo>∈</mo><mo>{</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo>}</mo></math></span>. In this article, we create a family of functions using repeated iterations of the function <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>a</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> and taking values of <span><math><mi>a</mi><mo>∈</mo><mo>{</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo>}</mo></math></span> in each iteration. Let <em>α</em> be an <em>n</em>-sequence of values for <em>a</em>, taken over <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo>}</mo></math></span>, and <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> be the resulting function. We present the form of <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> and use it to derive a closed formula for the number and length of cycles present in the functional graph of <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span>. We then determine the shape of the trees hanging from each cycle and gather all the results in our main theorem.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114500"},"PeriodicalIF":0.7,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143739294","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 oriented diameter of a bridgeless graph with the given path Pk 具有给定路径Pk的无桥图的定向直径

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2025-04-01 DOI: 10.1016/j.disc.2025.114509

Ruijuan Li, Shufeng Chen

引用次数: 0

O(VE) time algorithms for the Grundy (First-Fit) chromatic number of block graphs and graphs with large girth 块图和大周长图的Grundy （First-Fit）色数的O（VE）时间算法

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2025-03-27 DOI: 10.1016/j.disc.2025.114502

Manouchehr Zaker

{"title":"O(VE) time algorithms for the Grundy (First-Fit) chromatic number of block graphs and graphs with large girth","authors":"Manouchehr Zaker","doi":"10.1016/j.disc.2025.114502","DOIUrl":"10.1016/j.disc.2025.114502","url":null,"abstract":"<div><div>The Grundy (or First-Fit) chromatic number of a graph <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span>, denoted by <span><math><mi>Γ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> (or <span><math><msub><mrow><mi>χ</mi></mrow><mrow><msub><mrow></mrow><mrow><mi>FF</mi></mrow></msub></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>), is the maximum number of colors used by a First-Fit (greedy) coloring of <em>G</em>. The determining <span><math><mi>Γ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is <span>NP</span>-complete for various classes of graphs. Also there exists a constant <span><math><mi>c</mi><mo>></mo><mn>0</mn></math></span> such that the Grundy number is hard to approximate within the ratio <em>c</em>. We first obtain an <span><math><mi>O</mi><mo>(</mo><mi>V</mi><mi>E</mi><mo>)</mo></math></span> algorithm to determine the Grundy number of block graphs i.e. graphs in which every biconnected component is a complete graph. We prove that the Grundy number of a general graph <em>G</em> with cut-vertices is upper bounded by the Grundy number of a block graph corresponding to <em>G</em>. This provides a reasonable upper bound for the Grundy number of graphs with cut-vertices. Next, define <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>max</mi></mrow><mrow><mi>u</mi><mo>∈</mo><mi>V</mi></mrow></msub><mo>⁡</mo><mspace></mspace><msub><mrow><mi>max</mi></mrow><mrow><mi>v</mi><mo>∈</mo><mi>N</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>:</mo><mi>d</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>≤</mo><mi>d</mi><mo>(</mo><mi>u</mi><mo>)</mo></mrow></msub><mo>⁡</mo><mi>d</mi><mo>(</mo><mi>v</mi><mo>)</mo></math></span>. We obtain an <span><math><mi>O</mi><mo>(</mo><mi>V</mi><mi>E</mi><mo>)</mo></math></span> algorithm to determine <span><math><mi>Γ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> for graphs <em>G</em> whose girth <em>g</em> is at least <span><math><mn>2</mn><msub><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mn>1</mn></math></span>. This algorithm provides a polynomial time approximation algorithm within ratio <span><math><mi>min</mi><mo>⁡</mo><mo>{</mo><mn>1</mn><mo>,</mo><mo>(</mo><mi>g</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mo>(</mo><mn>2</mn><msub><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mn>2</mn><mo>)</mo><mo>}</mo></math></span> for <span><math><mi>Γ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of general graphs <em>G</em> with girth <em>g</em>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 9","pages":"Article 114502"},"PeriodicalIF":0.7,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143704983","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

Dense, irregular, yet always-graphic 3-uniform hypergraph degree sequences 密集的，不规则的，但总是-图形-均匀的超图度序列

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2025-03-27 DOI: 10.1016/j.disc.2025.114498

Runze Li , István Miklós

{"title":"Dense, irregular, yet always-graphic 3-uniform hypergraph degree sequences","authors":"Runze Li , István Miklós","doi":"10.1016/j.disc.2025.114498","DOIUrl":"10.1016/j.disc.2025.114498","url":null,"abstract":"<div><div>A 3-uniform hypergraph is a generalization of a simple graph where each hyperedge is a subset of exactly three vertices. The degree of a vertex in a hypergraph is the number of hyperedges incident with it. The degree sequence of a hypergraph is the sequence of the degrees of its vertices. The degree sequence problem for 3-uniform hypergraphs asks whether a 3-uniform hypergraph with a given degree sequence exists. Such a hypergraph is called a realization. Recently, Deza et al. proved that this problem is NP-complete. Although some special cases are simple, polynomial-time algorithms are only known for highly restricted degree sequences. The main result of our research is the following: if all degrees in a sequence <em>D</em> of length <em>n</em> are between <span><math><mfrac><mrow><mn>2</mn><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>63</mn></mrow></mfrac><mo>+</mo><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and <span><math><mfrac><mrow><mn>5</mn><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>63</mn></mrow></mfrac><mo>−</mo><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span>, the number of vertices is at least 45, and the degree sum is divisible by 3, then <em>D</em> has a 3-uniform hypergraph realization. Our proof is constructive, providing a polynomial-time algorithm for constructing such a hypergraph. To our knowledge, this is the first polynomial-time algorithm to construct a 3-uniform hypergraph realization of a highly irregular and dense degree sequence.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 9","pages":"Article 114498"},"PeriodicalIF":0.7,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143705055","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

Quaternary Legendre pairs II 第四纪勒让德对2

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2025-03-27 DOI: 10.1016/j.disc.2025.114501

Ilias S. Kotsireas , Christoph Koutschan , Arne Winterhof

{"title":"Quaternary Legendre pairs II","authors":"Ilias S. Kotsireas , Christoph Koutschan , Arne Winterhof","doi":"10.1016/j.disc.2025.114501","DOIUrl":"10.1016/j.disc.2025.114501","url":null,"abstract":"<div><div>Quaternary Legendre pairs are pertinent to the construction of quaternary Hadamard matrices and have many applications, for example in coding theory and communications.</div><div>In contrast to binary Legendre pairs, quaternary ones can exist for even length <em>ℓ</em> as well. It is conjectured that there is a quaternary Legendre pair for any even <em>ℓ</em>. The smallest open case until now had been <span><math><mi>ℓ</mi><mo>=</mo><mn>28</mn></math></span>, and <span><math><mi>ℓ</mi><mo>=</mo><mn>38</mn></math></span> was the only length <em>ℓ</em> with <span><math><mn>28</mn><mo>≤</mo><mi>ℓ</mi><mo>≤</mo><mn>60</mn></math></span> resolved before. Here we provide constructions for <span><math><mi>ℓ</mi><mo>=</mo><mn>28</mn><mo>,</mo><mn>30</mn><mo>,</mo><mn>32</mn></math></span>, and 34. In parallel and independently, Jedwab and Pender found a construction of quaternary Legendre pairs of length <span><math><mi>ℓ</mi><mo>=</mo><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn></math></span> for any prime power <span><math><mi>q</mi><mo>≡</mo><mn>1</mn><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mn>4</mn></math></span>, which in particular covers <span><math><mi>ℓ</mi><mo>=</mo><mn>30</mn></math></span>, 36, and 40, so that now <span><math><mi>ℓ</mi><mo>=</mo><mn>42</mn></math></span> is the smallest unresolved case.</div><div>The main new idea of this paper is a way to separate the search for the subsequences along even and odd indices which substantially reduces the complexity of the search algorithm.</div><div>In addition, we use Galois theory for cyclotomic fields to derive conditions which improve the PSD test.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 9","pages":"Article 114501"},"PeriodicalIF":0.7,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143705056","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

Improved upper bounds for six-valent integer distance graph coloring periods 改进的六价整数距离图着色周期上界

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2025-03-26 DOI: 10.1016/j.disc.2025.114496

Jonathan Cervantes , Mike Krebs

{"title":"Improved upper bounds for six-valent integer distance graph coloring periods","authors":"Jonathan Cervantes , Mike Krebs","doi":"10.1016/j.disc.2025.114496","DOIUrl":"10.1016/j.disc.2025.114496","url":null,"abstract":"<div><div>Given a set <em>S</em> of positive integers, the integer distance graph for <em>S</em> has the set of integers as its vertex set, where two vertices are adjacent if and only if the absolute value of their difference lies in <em>S</em>. In 2002, Zhu completely determined the chromatic number of integer distance graphs when <em>S</em> has cardinality 3, in which case the graphs have degree 6. Integer distance graphs can be defined equivalently as Cayley graphs on the group of integers under addition. In 1990 Eggleton, Erdős, and Skilton proved that if an integer distance graph of finite degree admits a proper <em>k</em>-coloring, then it admits a periodic proper <em>k</em>-coloring. They obtained an upper bound on the minimum such period but point out that it is quite large and very likely can be reduced considerably. In previous work, the authors of the present paper develop a general matrix method to approach the problem of finding chromatic numbers of abelian Cayley graphs. In this article we show that Zhu's theorem can be recovered as a special case of these results, and that in so doing we significantly improve the upper bounds on the periods of optimal colorings of these graphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114496"},"PeriodicalIF":0.7,"publicationDate":"2025-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143704715","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

Fractional colorings of partial t-trees with no large clique 无大团的部分t树的分式着色

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2025-03-25 DOI: 10.1016/j.disc.2025.114495

Peter Bradshaw

引用次数: 0

Tyshkevich's graph decomposition and the distinguishing numbers of unigraphs Tyshkevich的图分解和单图的区分数

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2025-03-24 DOI: 10.1016/j.disc.2025.114492

Christine T. Cheng

引用次数: 0