Discrete Mathematics最新文献

{"title":"Pursuit-evasion in graphs: Zombies, lazy zombies and a survivor","authors":"","doi":"10.1016/j.disc.2024.114220","DOIUrl":"10.1016/j.disc.2024.114220","url":null,"abstract":"<div><p>We study <em>zombies and survivor</em>, a variant of the game of cops and robber on graphs where the single survivor plays the role of the robber and attempts to escape from the zombies that play the role of the cops. The difference is that zombies must follow an edge of a shortest path towards the survivor on their turn. Let <span><math><mi>z</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the smallest number of zombies required to catch the survivor on a graph <em>G</em> with <em>n</em> vertices. We show that there exist outerplanar graphs and visibility graphs of simple polygons such that <span><math><mi>z</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>Θ</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span>. We also show that there exist maximum-degree-3 outerplanar graphs such that <span><math><mi>z</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>Ω</mi><mrow><mo>(</mo><mi>n</mi><mo>/</mo><mi>log</mi><mo>⁡</mo><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo></mrow></math></span>.</p><p>A zombie that can remain at its current vertex on its turn is called <em>lazy</em>. Let <span><math><msub><mrow><mi>z</mi></mrow><mrow><mi>L</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the smallest number of <em>lazy zombies</em> required to catch the survivor. The ability to remain at its current vertex on its turn makes lazy zombies more powerful than normal zombies but less powerful than cops. We prove that <span><math><msub><mrow><mi>z</mi></mrow><mrow><mi>L</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mn>2</mn></math></span> for connected outerplanar graphs which is tight in the worst case. We also show that in this case, the survivor is caught after <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> rounds. We then show that <span><math><msub><mrow><mi>z</mi></mrow><mrow><mi>L</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>k</mi></math></span> for connected graphs with treedepth <em>k</em> and that <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn><mi>k</mi></mrow></msup><mo>)</mo></math></span> rounds are sufficient to catch the survivor. The bound on treedepth implies that <span><math><msub><mrow><mi>z</mi></mrow><mrow><mi>L</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is at most <span><math><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo><mi>log</mi><mo>⁡</mo><mi>n</mi></math></span> for connected graphs with treewidth <em>k</em>, <span><math><mi>O</mi><mo>(</mo><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>)</mo></math></span> for connected planar graphs, <span><math><mi>O</mi><mo>(</mo><msqrt><mrow><mi>g</mi><mi>n</mi></mrow></msqrt><mo>)</mo></math></span> for connected graphs with genus <em>g</em> and <span><math><mi>O</mi><mo>(</mo><mi>h</mi><msqrt><mrow><mi>h</mi><mi>n</mi></mrow></msqrt><mo>)</mo></math></span> for connected graphs with any excluded <em>h</em>-vertex minor. Our results on lazy zombies still hold when an adversary chooses ","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003510/pdfft?md5=bb277c6c089d1c05a249b293a26a64fe&pid=1-s2.0-S0012365X24003510-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142077548","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}

{"title":"Fractional revival on Cayley graphs over abelian groups","authors":"","doi":"10.1016/j.disc.2024.114218","DOIUrl":"10.1016/j.disc.2024.114218","url":null,"abstract":"<div><p>In this paper, we investigate the existence of fractional revival on Cayley graphs over finite abelian groups. We give a necessary and sufficient condition for Cayley graphs over finite abelian groups to have fractional revival. As applications, the existence of fractional revival on circulant graphs and cubelike graphs are characterized.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142040369","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 flag-transitive, point-primitive 2-designs from complete graphs","authors":"","doi":"10.1016/j.disc.2024.114217","DOIUrl":"10.1016/j.disc.2024.114217","url":null,"abstract":"<div><p>In this paper, we study 2-designs <span><math><mi>D</mi><mo>=</mo><mo>(</mo><mi>P</mi><mo>,</mo><msup><mrow><mi>B</mi></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msup><mo>)</mo></math></span>, where <span><math><mi>P</mi></math></span> can be viewed as the edge set of the complete graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, and <em>B</em> can be identified as the edge set of a subgraph of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. We give a necessary condition for <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> to be flag-transitive, and then present three ways to construct such 2-designs admitting a flag-transitive, point-primitive automorphism group <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. As an application, all pairs <span><math><mo>(</mo><mi>D</mi><mo>,</mo><mi>G</mi><mo>)</mo></math></span> are determined, where <span><math><mi>D</mi></math></span> is a 2-<span><math><mo>(</mo><mi>v</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math></span> design with <span><math><mi>gcd</mi><mo>⁡</mo><mo>(</mo><mi>v</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>=</mo><mn>3</mn></math></span> or 4, and <em>G</em> is flag-transitive with <span><math><mi>S</mi><mi>o</mi><mi>c</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> for <span><math><mi>n</mi><mo>≥</mo><mn>5</mn></math></span>. Furthermore, we show that there are infinite flag-transitive, point-primitive 2-<span><math><mo>(</mo><mi>v</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math></span> designs with <span><math><mi>gcd</mi><mo>⁡</mo><mo>(</mo><mi>v</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>≤</mo><msup><mrow><mo>(</mo><mi>v</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></math></span> and alternating socle <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with <span><math><mi>v</mi><mo>=</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow></math></span>.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003480/pdfft?md5=8dd45f6c8de1e9aaa5ee26ce47fc990b&pid=1-s2.0-S0012365X24003480-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142020505","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}

{"title":"Partition theorems and the Chinese Remainder Theorem","authors":"","doi":"10.1016/j.disc.2024.114221","DOIUrl":"10.1016/j.disc.2024.114221","url":null,"abstract":"<div><p>The famous partition theorem of Euler states that partitions of <em>n</em> into distinct parts are equinumerous with partitions of <em>n</em> into odd parts. Another famous partition theorem due to MacMahon states that the number of partitions of <em>n</em> with all parts repeated at least once equals the number of partitions of <em>n</em> where all parts must be even or congruent to <span><math><mn>3</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>6</mn><mo>)</mo></math></span>. These partition theorems were further extended by Glaisher, Andrews, Subbarao, Nyirenda and Mugwangwavari. In this paper, we utilize the Chinese Remainder Theorem to prove a comprehensive partition theorem that encompasses all existing partition theorems. We also give a natural generalization of Euler's theorem based on a special complete residue system. Furthermore, we establish interesting congruence connections between the partition function <span><math><mi>p</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and related partition functions.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142013003","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":"Online size Ramsey numbers: Path vs C4","authors":"","doi":"10.1016/j.disc.2024.114214","DOIUrl":"10.1016/j.disc.2024.114214","url":null,"abstract":"<div><p>Given two graphs <em>G</em> and <em>H</em>, a size Ramsey game is played on the edge set of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>N</mi></mrow></msub></math></span>. In every round, Builder selects an edge and Painter colours it red or blue. Builder's goal is to force Painter to create a red copy of <em>G</em> or a blue copy of <em>H</em> as soon as possible. The online (size) Ramsey number <span><math><mover><mrow><mi>r</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> is the number of rounds in the game provided Builder and Painter play optimally. We prove that <span><math><mover><mrow><mi>r</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>≤</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></math></span> for every <span><math><mi>n</mi><mo>≥</mo><mn>8</mn></math></span>. The upper bound matches the lower bound obtained by J. Cyman, T. Dzido, J. Lapinskas, and A. Lo, so we get <span><math><mover><mrow><mi>r</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>=</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></math></span> for <span><math><mi>n</mi><mo>≥</mo><mn>8</mn></math></span>. Our proof for <span><math><mi>n</mi><mo>≤</mo><mn>13</mn></math></span> is computer-assisted. The bound <span><math><mover><mrow><mi>r</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>≤</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></math></span> solves also the “all cycles vs. <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>” game for <span><math><mi>n</mi><mo>≥</mo><mn>8</mn></math></span> – it implies that it takes Builder <span><math><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></math></span> rounds to force Painter to create a blue path on <em>n</em> vertices or any red cycle.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142013002","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":"Further results on large sets plus of partitioned incomplete Latin squares","authors":"","doi":"10.1016/j.disc.2024.114215","DOIUrl":"10.1016/j.disc.2024.114215","url":null,"abstract":"<div><p>In this paper, we continue to study the existence of large sets plus of partitioned incomplete Latin squares of type <span><math><msup><mrow><mi>g</mi></mrow><mrow><mi>n</mi></mrow></msup><msup><mrow><mo>(</mo><mi>u</mi><mi>g</mi><mo>)</mo></mrow><mrow><mn>1</mn></mrow></msup></math></span>, denoted by LSPILS<span><math><msup><mrow></mrow><mrow><mo>+</mo></mrow></msup><mo>(</mo><msup><mrow><mi>g</mi></mrow><mrow><mi>n</mi></mrow></msup><msup><mrow><mo>(</mo><mi>u</mi><mi>g</mi><mo>)</mo></mrow><mrow><mn>1</mn></mrow></msup><mo>)</mo></math></span>. We almost solve the existence of an LSPILS<span><math><msup><mrow></mrow><mrow><mo>+</mo></mrow></msup><mo>(</mo><msup><mrow><mi>g</mi></mrow><mrow><mi>n</mi></mrow></msup><msup><mrow><mo>(</mo><mi>u</mi><mi>g</mi><mo>)</mo></mrow><mrow><mn>1</mn></mrow></msup><mo>)</mo></math></span> for any integer <span><math><mi>g</mi><mo>≥</mo><mn>1</mn></math></span> and <span><math><mi>u</mi><mo>=</mo><mn>3</mn><mo>,</mo><mn>4</mn></math></span> with some possible exceptions.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003467/pdfft?md5=a7bdcbba4d2ea5621caf6949ac6fa294&pid=1-s2.0-S0012365X24003467-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142006835","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}

{"title":"Turán numbers of general star forests in hypergraphs","authors":"","doi":"10.1016/j.disc.2024.114219","DOIUrl":"10.1016/j.disc.2024.114219","url":null,"abstract":"<div><p>Let <span><math><mi>F</mi></math></span> be a family of <em>r</em>-uniform hypergraphs, and let <em>H</em> be an <em>r</em>-uniform hypergraph. Then <em>H</em> is called <span><math><mi>F</mi></math></span>-free if it does not contain any member of <span><math><mi>F</mi></math></span> as a subhypergraph. The Turán number of <span><math><mi>F</mi></math></span>, denoted by <span><math><msub><mrow><mi>ex</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span>, is the maximum number of hyperedges in an <span><math><mi>F</mi></math></span>-free <em>n</em>-vertex <em>r</em>-uniform hypergraph. Our current results are motivated by earlier results on Turán numbers of star forests and hypergraph star forests. In particular, Lidický et al. (2013) <span><span>[17]</span></span> determined the Turán number <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> of a star forest <em>F</em> for sufficiently large <em>n</em>. Recently, Khormali and Palmer (2022) <span><span>[13]</span></span> generalized the above result to three different well-studied hypergraph settings (the expansions of a graph, linear hypergraphs and Berge hypergraphs), but restricted to the case that all stars in the hypergraph star forests are identical. We further generalize these results to general star forests in hypergraphs.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003509/pdfft?md5=f55a8417dd66a400951a48477694c9f9&pid=1-s2.0-S0012365X24003509-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142006836","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}

{"title":"The VC dimension of quadratic residues in finite fields","authors":"","doi":"10.1016/j.disc.2024.114192","DOIUrl":"10.1016/j.disc.2024.114192","url":null,"abstract":"<div><p>We study the Vapnik–Chervonenkis (VC) dimension of the set of quadratic residues (i.e. squares) in finite fields, <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>, when considered as a subset of the additive group. We conjecture that as <span><math><mi>q</mi><mo>→</mo><mo>∞</mo></math></span>, the squares have the maximum possible VC-dimension, viz. <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><msub><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⁡</mo><mi>q</mi></math></span>. We prove, using the Weil bound for multiplicative character sums, that the VC-dimension is <span><math><mo>⩾</mo><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><msub><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⁡</mo><mi>q</mi></math></span>. We also provide numerical evidence for our conjectures. The results generalize to multiplicative subgroups <span><math><mi>Γ</mi><mo>⊆</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mo>×</mo></mrow></msubsup></math></span> of bounded index.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003236/pdfft?md5=cb2593a83f33c425a70d3257432c949e&pid=1-s2.0-S0012365X24003236-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141993205","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}

{"title":"On the direct and inverse zero-sum problems over non-split metacyclic group","authors":"","doi":"10.1016/j.disc.2024.114213","DOIUrl":"10.1016/j.disc.2024.114213","url":null,"abstract":"<div><p>Let <span><math><mi>G</mi><mo>=</mo><mrow><mo>〈</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo>|</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>=</mo><mn>1</mn><mo>,</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><mi>x</mi><mi>y</mi><msup><mrow><mi>x</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><msup><mrow><mi>y</mi></mrow><mrow><mi>ℓ</mi></mrow></msup><mo>〉</mo></mrow></math></span> be the non-split metacyclic group with <span><math><msup><mrow><mi>ℓ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>≡</mo><mn>1</mn><mspace></mspace><mo>(</mo><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mspace></mspace><mn>2</mn><mi>n</mi><mo>)</mo></math></span> and <span><math><mi>ℓ</mi><mo>≢</mo><mo>±</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn><mspace></mspace><mo>(</mo><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mspace></mspace><mn>2</mn><mi>n</mi><mo>)</mo></math></span>. In this paper, we obtain the exact values of small Davenport constant <span><math><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, Gao constant <span><math><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, <em>η</em>-constant <span><math><mi>η</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and Erdős-Ginzburg-Ziv constant <span><math><mi>s</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. Additionally, we study the associated inverse problems on <span><math><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, <span><math><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, <span><math><mi>η</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and <span><math><mi>s</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. In 2003, Gao conjectured that <span><math><mi>s</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>η</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mtext>exp</mtext><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span> for any finite group <em>G</em>. In 2005, Gao and Zhuang conjectured that <span><math><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mo>|</mo><mi>G</mi><mo>|</mo></math></span> for any finite group <em>G</em>. As a result, we confirm the two conjectures for non-split metacyclic groups.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141991343","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":"Cut-down de Bruijn sequences","authors":"","doi":"10.1016/j.disc.2024.114204","DOIUrl":"10.1016/j.disc.2024.114204","url":null,"abstract":"<div><p>A cut-down de Bruijn sequence is a cyclic string of length <em>L</em>, where <span><math><mn>1</mn><mo>≤</mo><mi>L</mi><mo>≤</mo><msup><mrow><mi>k</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, such that every substring of length <em>n</em> appears <em>at most</em> once. Etzion [<em>Theor. Comp. Sci</em> 44 (1986)] introduced an algorithm to construct binary cut-down de Bruijn sequences requiring <span><math><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> simple <em>n</em>-bit operations per symbol generated. In this paper, we simplify the algorithm and improve the running time to <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> time per symbol generated using <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> space. Additionally, we develop the first successor-rule approach for constructing a binary cut-down de Bruijn sequence by leveraging recent ranking/unranking algorithms for fixed-density Lyndon words. Finally, we develop an algorithm to generate cut-down de Bruijn sequences for <span><math><mi>k</mi><mo>&gt;</mo><mn>2</mn></math></span> that runs in <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> time per symbol using <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> space after some initialization.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003352/pdfft?md5=dc65cfb8e32bb465a8c99176a8b278b0&pid=1-s2.0-S0012365X24003352-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141984720","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}