Discrete Mathematics最新文献_第2页

The oriented diameter of a bridgeless graph with the given path Pk

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

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

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

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

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

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

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

The king degree and the second out-degree of tournaments

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2025-03-21 DOI: 10.1016/j.disc.2025.114497

Aya Alhussein , Ayman El Zein

{"title":"The king degree and the second out-degree of tournaments","authors":"Aya Alhussein , Ayman El Zein","doi":"10.1016/j.disc.2025.114497","DOIUrl":"10.1016/j.disc.2025.114497","url":null,"abstract":"<div><div>In a digraph, the second out-degree of a vertex <em>x</em>, denoted by <span><math><msup><mrow><mi>d</mi></mrow><mrow><mo>+</mo><mo>+</mo></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo></math></span>, is the number of vertices <em>y</em> such that <span><math><msup><mrow><mi>d</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>=</mo><mn>2</mn></math></span>, where <span><math><msup><mrow><mi>d</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> is the length of the shortest <em>xy</em>-directed path, if it exists. It is obvious that the sum of the first out-degrees of the vertices in a digraph is nothing but the number of its arcs. Unlike the first out-degree, the summation of the second out-degrees of the vertices in a digraph is not constant with respect to the number of vertices and arcs. In this paper, we characterize, as a function of some integer <em>n</em>, the values that can be the summation of the second out-degrees of the vertices in a tournament of order <em>n</em>. Throughout the paper, we use the new concept of king degree in order to settle the problem. The king degree of a vertex <em>x</em> is the number of vertices that can be reached from <em>x</em> by a directed path of length at most 2. Several open problems are introduced in the last section of the paper.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 9","pages":"Article 114497"},"PeriodicalIF":0.7,"publicationDate":"2025-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143683710","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

Combinatorics on words and generating Dirichlet series of automatic sequences

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2025-03-21 DOI: 10.1016/j.disc.2025.114487

Jean-Paul Allouche , Jeffrey Shallit , Manon Stipulanti

{"title":"Combinatorics on words and generating Dirichlet series of automatic sequences","authors":"Jean-Paul Allouche , Jeffrey Shallit , Manon Stipulanti","doi":"10.1016/j.disc.2025.114487","DOIUrl":"10.1016/j.disc.2025.114487","url":null,"abstract":"<div><div>Generating series are crucial in enumerative combinatorics, analytic combinatorics, and combinatorics on words. Though it might seem at first view that generating Dirichlet series are less used in these fields than ordinary and exponential generating series, there are many notable papers where they play a fundamental role, as can be seen in particular in the work of Flajolet and several of his co-authors. In this paper, we study Dirichlet series of integers with missing digits or blocks of digits in some integer base <em>b</em>; i.e., where the summation ranges over the integers whose expansions form some language strictly included in the set of all words over the alphabet <span><math><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>b</mi><mo>−</mo><mn>1</mn><mo>}</mo></math></span> that do not begin with a 0. We show how to unify and extend results proved by Nathanson in 2021 and by Köhler and Spilker in 2009. En route, we encounter several sequences from Sloane's On-Line Encyclopedia of Integer Sequences, as well as some famous <em>b</em>-automatic sequences or <em>b</em>-regular sequences. We also consider a specific sequence that is not <em>b</em>-regular.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114487"},"PeriodicalIF":0.7,"publicationDate":"2025-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143684325","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

Sign-twisted generating functions of the odd length for Weyl groups of type D

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2025-03-21 DOI: 10.1016/j.disc.2025.114494

Haihang Gu , Houyi Yu

引用次数: 0