Discrete Mathematics最新文献

{"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}

{"title":"A survey of complex generalized weighing matrices and a construction of quantum error-correcting codes","authors":"","doi":"10.1016/j.disc.2024.114201","DOIUrl":"10.1016/j.disc.2024.114201","url":null,"abstract":"<div><p>Some combinatorial designs, such as Hadamard matrices, have been extensively researched and are familiar to readers across the spectrum of Science and Engineering. They arise in diverse fields such as cryptography, communication theory, and quantum computing. Objects like this also lend themselves to compelling mathematics problems, such as the Hadamard conjecture. However, complex generalized weighing matrices, which generalize Hadamard matrices, have not received anything like the same level of scrutiny. Motivated by an application to the construction of quantum error-correcting codes, which we outline in the latter sections of this paper, we survey the existing literature on complex generalized weighing matrices. We discuss and extend upon the known existence conditions and constructions, and compile known existence results for small parameters. Using these matrices we construct Hermitian self orthogonal codes over finite fields of square order, and consequently some interesting quantum codes are constructed to demonstrate the value of complex generalized weighing matrices.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003327/pdfft?md5=529c4d63c13ac71c4519138cdb73c99c&pid=1-s2.0-S0012365X24003327-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141978438","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 Terwilliger algebras of Odd graphs and Doubled Odd graphs","authors":"","doi":"10.1016/j.disc.2024.114216","DOIUrl":"10.1016/j.disc.2024.114216","url":null,"abstract":"<div><p>For an integer <span><math><mi>m</mi><mo>≥</mo><mn>1</mn></math></span>, let <span><math><mi>S</mi><mo>=</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>2</mn><mi>m</mi><mo>+</mo><mn>1</mn><mo>}</mo></math></span>. Denote by <span><math><mn>2</mn><mo>.</mo><msub><mrow><mi>O</mi></mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> the Doubled Odd graph on <em>S</em> with vertex set <span><math><mi>X</mi><mo>:</mo><mo>=</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>S</mi></mtd></mtr><mtr><mtd><mi>m</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>∪</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>S</mi></mtd></mtr><mtr><mtd><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow></math></span>. By folding this graph, one can obtain a new graph called Odd graph <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> with vertex set <span><math><mi>X</mi><mo>:</mo><mo>=</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>S</mi></mtd></mtr><mtr><mtd><mi>m</mi></mtd></mtr></mtable><mo>)</mo></mrow></math></span>. In this paper, we shall study the Terwilliger algebras of <span><math><mn>2</mn><mo>.</mo><msub><mrow><mi>O</mi></mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>. We first consider the case of <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>. With respect to any fixed vertex <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><mi>X</mi></math></span>, let <span><math><mi>A</mi><mo>:</mo><mo>=</mo><mi>A</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math></span> denote the centralizer algebra of the stabilizer of <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> in the automorphism group of <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>, and <span><math><mi>T</mi><mo>:</mo><mo>=</mo><mi>T</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math></span> the Terwilliger algebra of <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>. For the algebras <span><math><mi>A</mi></math></span> and <span><math><mi>T</mi></math></span>: (i) we construct a basis of <span><math><mi>A</mi></math></span> by the stabilizer of <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> acting on <span><math><mi>X</mi><mo>×</mo><mi>X</mi></math></span>, compute its dimension and show that <span><math><mi>A</mi><mo>=</mo><mi>T</mi></math></span>; (ii) for <span><math><mi>m</mi><mo>≥</mo><mn>3</mn></math></span>, we give all the isomorphism classes of irreducible <span><math><mi>T</mi></math></span>-modu","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003479/pdfft?md5=8c465dc78658321c3a6c455f5d3877fe&pid=1-s2.0-S0012365X24003479-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141978439","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":"New methods for constructing AEAQEC codes","authors":"","doi":"10.1016/j.disc.2024.114202","DOIUrl":"10.1016/j.disc.2024.114202","url":null,"abstract":"<div><p>Recently, Liu and Liu gave the Singleton bound for pure asymmetric entanglement-assisted quantum error-correcting (AEAQEC) codes. They constructed three new families of AQEAEC codes by means of Vandermonde matrices, generalized Reed-Solomon (GRS) codes and cyclic codes. In this paper, we first exhibit the Singleton bound for any AEAQEC codes. Then we construct AEAQEC codes by two distinct constacyclic codes. By means of repeated-root cyclic codes, we construct new AEAQEC MDS codes. In addition, our methods allow for easily calculating the dimensions, <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>z</mi></mrow></msub></math></span>, <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>x</mi></mrow></msub></math></span> and the number <em>c</em> of pre-shared maximally entangled states of AEAQEC codes.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141978322","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":"Ramsey numbers and a general Erdős-Rogers function","authors":"","doi":"10.1016/j.disc.2024.114203","DOIUrl":"10.1016/j.disc.2024.114203","url":null,"abstract":"<div><p>Given a graph <em>F</em>, let <span><math><mi>L</mi><mo>(</mo><mi>F</mi><mo>)</mo></math></span> be a fixed finite family of graphs consisting of a <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> and some bipartite graphs relying on an <em>s</em>-partite subgraph partitioning of edges of <em>F</em>. Define <span><math><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></math></span>-graph by an <span><math><mi>m</mi><mo>×</mo><mi>n</mi></math></span> bipartite graph with <span><math><mi>n</mi><mo>≥</mo><mi>m</mi></math></span> such that all vertices in the part of size <em>n</em> have degree <em>a</em> and all vertices in the part of size <em>m</em> have degree <span><math><mi>b</mi><mo>≥</mo><mi>a</mi></math></span>. In this paper, building upon the work of Janzer and Sudakov (2023⁺) and combining with the idea of Conlon, Mattheus, Mubayi and Verstraëte (2023⁺) we obtain that for each <span><math><mi>s</mi><mo>≥</mo><mn>2</mn></math></span>, if there exists an <span><math><mi>L</mi><mo>(</mo><mi>F</mi><mo>)</mo></math></span>-free <span><math><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></math></span>-graph, then there exists an <em>F</em>-free graph <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> with at least <span><math><mi>n</mi><msup><mrow><mi>a</mi></mrow><mrow><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>s</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msup><mo>−</mo><mn>1</mn></math></span> vertices in which every vertex subset of size <span><math><mi>m</mi><msup><mrow><mi>a</mi></mrow><mrow><mo>−</mo><mfrac><mrow><mi>s</mi></mrow><mrow><mi>s</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msup><msup><mrow><mi>log</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>⁡</mo><mo>(</mo><mi>a</mi><mi>n</mi><mo>)</mo></math></span> contains a copy of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span>. As applications, we obtain some upper bounds of general Erdős-Rogers functions for some special graphs of <em>F</em>. Moreover, we obtain the multicolor Ramsey numbers <span><math><msub><mrow><mi>r</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>;</mo><mi>t</mi><mo>)</mo><mo>=</mo><mover><mrow><mi>Ω</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><msup><mrow><mi>t</mi></mrow><mrow><mfrac><mrow><mn>3</mn><mi>k</mi></mrow><mrow><mn>7</mn></mrow></mfrac><mo>+</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> and <span><math><msub><mrow><mi>r</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>7</mn></mrow></msub><mo>;</mo><mi>t</mi><mo>)</mo><mo>=</mo><mover><mrow><mi>Ω</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><msup><mrow><mi>t</mi></mrow><mrow><mfrac><mrow><mi>k</mi></mr","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141978323","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":"A complete classification of edge-primitive graphs of valency 6","authors":"","doi":"10.1016/j.disc.2024.114205","DOIUrl":"10.1016/j.disc.2024.114205","url":null,"abstract":"<div><p>In 2020, the first author and Pan proved that every edge-primitive graph of valency 6 is 2-arc-transitive, and except the complete bipartite graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>6</mn><mo>,</mo><mn>6</mn></mrow></msub></math></span>, the automorphism group is almost simple, and they determined such graphs having a solvable edge stabilizer. The nonsolvable edge stabilizer case is settled in this work, which leads to a complete classification of edge-primitive graphs of valency 6.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141978182","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}