Discrete Mathematics最新文献

{"title":"Every even cycle of order at least 8 has a mirror labeling","authors":"Jonathan Calzadillas ,&nbsp;Dan McQuillan ,&nbsp;James M. McQuillan","doi":"10.1016/j.disc.2025.114503","DOIUrl":"10.1016/j.disc.2025.114503","url":null,"abstract":"<div><div>A mirror labeling of the cycle <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is a vertex-magic total labeling (VMTL) for the cycle <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with the property that if <em>x</em> is a vertex label, then <span><math><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>−</mo><mi>x</mi></math></span> is an edge label, for each <span><math><mn>1</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>2</mn><mi>n</mi></math></span>. (Note that any mirror labeling for <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> can be easily converted into an edge-magic total labeling for <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with the same property, and vice versa.) It has been known for decades that every odd cycle has a mirror labeling. Mirror labelings for <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with <em>n</em> even are considerably more difficult to construct generally, with only the case <span><math><mi>n</mi><mo>≡</mo><mn>2</mn></math></span> mod 8 having been provided. In this paper, we obtain mirror labelings for all remaining cases, namely <span><math><mi>n</mi><mo>≡</mo><mn>6</mn></math></span> mod 8, <span><math><mi>n</mi><mo>≥</mo><mn>14</mn></math></span> and <span><math><mi>n</mi><mo>≡</mo><mn>0</mn></math></span> mod 4, <span><math><mi>n</mi><mo>≥</mo><mn>8</mn></math></span>.</div><div>This result has significant ramifications for the study of vertex-magic total labelings of graphs generally. A quarter century ago, James MacDougall provided his guiding conjecture positing that every regular graph of degree at least 2 has a VMTL, except for the disjoint union <span><math><mn>2</mn><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>. Ian Gray showed that every Hamiltonian regular graph of odd order possesses a VMTL, and introduced mirror vertex-magic total labelings as a tool to obtain a similar, general result for even order regular graphs. However, a key technical part of his program was missing, namely, the existence of mirror VMTLs for even order cycles. A mirror labeling is a particular kind of mirror VMTL. Thus, the results of this work provide the missing piece required for Gray's program. It now follows, that any Hamiltonian <span><math><mo>(</mo><mn>4</mn><mi>t</mi><mo>+</mo><mn>2</mn><mo>)</mo></math></span>-regular graph of any even order (<span><math><mi>n</mi><mo>≥</mo><mn>8</mn></math></span>, <span><math><mi>t</mi><mo>≥</mo><mn>0</mn></math></span>) must have a VMTL. This provides substantial new progress towards resolving MacDougall's Conjecture.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114503"},"PeriodicalIF":0.7,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761184","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}

{"title":"Rainbow directed version of Dirac's theorem","authors":"Hao Li ,&nbsp;Luyi Li ,&nbsp;Ping Li ,&nbsp;Xueliang Li","doi":"10.1016/j.disc.2025.114506","DOIUrl":"10.1016/j.disc.2025.114506","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>:</mo><mi>i</mi><mo>∈</mo><mo>[</mo><mi>s</mi><mo>]</mo><mo>}</mo></math></span> be a collection of not necessarily distinct graphs on the same vertex set <em>V</em>. A graph <em>H</em> is called <em>rainbow</em> in <span><math><mi>G</mi></math></span> if any two edges of <em>H</em> belong to different graphs of <span><math><mi>G</mi></math></span>. In 2020, Joos and Kim proved a rainbow version of Dirac's theorem. In this paper, we prove a rainbow directed version of Dirac's theorem asymptotically: For each <span><math><mn>0</mn><mo>&lt;</mo><mi>ε</mi><mo>&lt;</mo><mn>1</mn></math></span>, there exists an integer <em>N</em> such that when <span><math><mi>n</mi><mo>≥</mo><mi>N</mi></math></span> the following holds. Let <span><math><mi>D</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>:</mo><mi>i</mi><mo>∈</mo><mo>[</mo><mi>n</mi><mo>]</mo><mo>}</mo></math></span> be a collection of <em>n</em>-vertex digraphs on the same vertex set <em>V</em>. If both the out-degree and the in-degree of <em>v</em> are at least <span><math><mrow><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>ε</mi><mo>)</mo></mrow><mi>n</mi></math></span> for each vertex <span><math><mi>v</mi><mo>∈</mo><mi>V</mi></math></span> and each integer <span><math><mi>i</mi><mo>∈</mo><mo>[</mo><mi>n</mi><mo>]</mo></math></span>, then <span><math><mi>D</mi></math></span> contains a rainbow Hamiltonian cycle. Furthermore, we provide a sufficient condition for the existence of arbitrary rainbow tournaments in a collection of <em>n</em>-vertex digraphs, where a <em>tournament</em> is an oriented graph of a complete graph.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114506"},"PeriodicalIF":0.7,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761335","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}

{"title":"Constructing balanced 2p-variable rotation symmetric Boolean functions with optimal algebraic immunity, high nonlinearity and high algebraic degree","authors":"Jiao Du ,&nbsp;Xiaoting Chen ,&nbsp;Yongxia Mao ,&nbsp;Qiang Gao ,&nbsp;Tianyin Wang","doi":"10.1016/j.disc.2025.114513","DOIUrl":"10.1016/j.disc.2025.114513","url":null,"abstract":"<div><div>How to design cryptographic Boolean functions is a challenge work in the design of stream and block ciphers. Cryptographic criteria of Boolean functions are connected with some known cryptanalytic attacks. To resist these known attacks, it is important to search Boolean functions with some properties, including balancedness, optimal algebraic immunity, high algebraic degree, good nonlinearity, high correlation immunity, etc. Rotation symmetric Boolean functions (RSBFs) can have these properties simultaneously. In this paper, we propose a new class of balanced 2<em>p</em>-variable RSBFs based on the compositions of an integer, where <em>p</em> is an odd prime. It is found that the functions of this class have optimal algebraic immunity, and their nonlinearity reaches <span><math><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mn>2</mn><mi>p</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mi>p</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mn>2</mn><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>3</mn></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msubsup><mrow><mo>(</mo><mi>i</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>i</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>p</mi><mo>−</mo><mi>i</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>η</mi></mrow></msub><mo>+</mo><mn>1</mn></math></span> (where <span><math><msub><mrow><mi>N</mi></mrow><mrow><mi>η</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>p</mi><mo>−</mo><mn>2</mn><mo>−</mo><mrow><mo>(</mo><mi>p</mi><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mn>4</mn><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> and <em>p</em> is an odd prime), which is higher than the previously constructed balanced even-variable RSBFs with optimal algebraic immunity. At the same time, the algebraic degree of the constructed functions are studied, and the results show that they can be optimal under certain conditions.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114513"},"PeriodicalIF":0.7,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761336","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}

{"title":"Complete weight enumerators and weight hierarchies of two classes of linear codes","authors":"Jiawei He,&nbsp;Yinjin Liao","doi":"10.1016/j.disc.2025.114510","DOIUrl":"10.1016/j.disc.2025.114510","url":null,"abstract":"<div><div>The study of generalized Hamming weights for linear coding is an important area of research in coding theory as it provides valuable structural information about coding and plays a crucial role in determining the performance of coding in various applications. In this paper, two distinct classes of linear codes are devised through the selection of two particular defining sets. Initially, the weight distributions of these codes are ascertained. Subsequently, by conducting a detailed analysis of the intersections between the defining sets and the duals of all <em>r</em>-dimensional subspaces, the complete weight hierarchies of the two classes of linear codes are successfully determined.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 9","pages":"Article 114510"},"PeriodicalIF":0.7,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143759598","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}

{"title":"Quasi-orthogonal extension of symmetric matrices","authors":"Abderrahim Boussaïri ,&nbsp;Brahim Chergui ,&nbsp;Zaineb Sarir ,&nbsp;Mohamed Zouagui","doi":"10.1016/j.disc.2025.114517","DOIUrl":"10.1016/j.disc.2025.114517","url":null,"abstract":"<div><div>An <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> real matrix <em>Q</em> is quasi-orthogonal if <span><math><msup><mrow><mi>Q</mi></mrow><mrow><mo>⊤</mo></mrow></msup><mi>Q</mi><mo>=</mo><mi>q</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> for some positive real number <em>q</em>. If <em>M</em> is a principal sub-matrix of a quasi-orthogonal matrix <em>Q</em>, we say that <em>Q</em> is a quasi-orthogonal extension of <em>M</em>. In a recent work, the authors have investigated this notion for the class of real skew-symmetric matrices. Using a different approach, this paper addresses the case of symmetric matrices.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114517"},"PeriodicalIF":0.7,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761337","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}

{"title":"On proper hamiltonicity and proper (even) pancyclicity of arc-colored complete (balanced bipartite) digraphs","authors":"Mengyu Duan ,&nbsp;Zhiwei Guo ,&nbsp;Binlong Li ,&nbsp;Shenggui Zhang","doi":"10.1016/j.disc.2025.114507","DOIUrl":"10.1016/j.disc.2025.114507","url":null,"abstract":"<div><div>A subdigraph of an arc-colored digraph is called properly colored if its every pair of consecutive arcs have distinct colors. We call an arc-colored digraph <em>D</em> properly hamiltonian if it contains a properly colored Hamilton cycle, and properly (even) pancyclic if it contains a properly colored cycle of length <em>k</em> for every (even) <em>k</em> with <span><math><mn>2</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mo>|</mo><mi>V</mi><mo>(</mo><mi>D</mi><mo>)</mo><mo>|</mo></math></span>. In this paper, we first obtain some color number conditions for the existence of properly colored Hamilton cycles of arc-colored complete (balanced bipartite) digraphs, and further prove that the these conditions can still guarantee the (even) pancyclicity of arc-colored complete (balanced bipartite) digraphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114507"},"PeriodicalIF":0.7,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761182","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}

{"title":"The graph of a family of functions over quadratic extensions of finite fields","authors":"Claude Gravel ,&nbsp;Daniel Panario ,&nbsp;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}

{"title":"The oriented diameter of a bridgeless graph with the given path Pk","authors":"Ruijuan Li,&nbsp;Shufeng Chen","doi":"10.1016/j.disc.2025.114509","DOIUrl":"10.1016/j.disc.2025.114509","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span> be a bridgeless undirected graph. The oriented diameter of <em>G</em> is the minimum diameter of any strongly connected orientation of <em>G</em>. Dankelmann, Guo and Surmacs [J. Graph Theory, 88 (2018), 5-17] showed that every bridgeless graph <em>G</em> of order <em>n</em> has an oriented diameter at most <span><math><mi>n</mi><mo>−</mo><mi>Δ</mi><mo>+</mo><mn>3</mn></math></span>, where Δ is the maximum degree of <em>G</em>. By defining <span><math><msub><mrow><mi>N</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>H</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>⋃</mo></mrow><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>H</mi><mo>)</mo></mrow></msub><msub><mrow><mi>N</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo><mo>∖</mo><mi>V</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> for every subgraph <em>H</em> of <em>G</em>, they proved that for an edge <em>e</em>, <em>G</em> has an orientation of diameter at most <span><math><mi>n</mi><mo>−</mo><mo>|</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>e</mi><mo>)</mo><mo>|</mo><mo>+</mo><mn>5</mn></math></span>. In this paper, we generalize the above-mentioned results by substituting a vertex or an edge <em>e</em> by a given path <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> in <em>G</em>. We first give an algorithm to cover <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> with some specific cycles, and then prove the upper bound <span><math><mi>n</mi><mo>−</mo><mo>|</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo><mo>|</mo><mo>+</mo><mn>2</mn><mo>⌊</mo><mfrac><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌋</mo><mo>+</mo><mn>3</mn></math></span> on the oriented diameter. We provide examples to show that our bound is sharp.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 9","pages":"Article 114509"},"PeriodicalIF":0.7,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143737820","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}

{"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>&gt;</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}

{"title":"Dense, irregular, yet always-graphic 3-uniform hypergraph degree sequences","authors":"Runze Li ,&nbsp;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}