{"title":"Group irregularity strength of disconnected graphs","authors":"Sylwia Cichacz, Barbara Krupińska","doi":"10.1016/j.disc.2025.114548","DOIUrl":"10.1016/j.disc.2025.114548","url":null,"abstract":"<div><div>We investigate the <em>group irregular strength</em> <span><math><mo>(</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span> of graphs, i.e. the smallest value of <em>s</em> such that for any Abelian group Γ of order <em>s</em> exists a function <span><math><mi>g</mi><mo>:</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>→</mo><mi>Γ</mi></math></span> such that sums of edge labels at every vertex is distinct. We give results for bound and exact values of <span><math><mo>(</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span> for graphs without small stars as components.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 9","pages":"Article 114548"},"PeriodicalIF":0.7,"publicationDate":"2025-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143902114","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":"Optimal coloring of (P2 + P3, gem)-free graphs","authors":"Arnab Char, T. Karthick","doi":"10.1016/j.disc.2025.114554","DOIUrl":"10.1016/j.disc.2025.114554","url":null,"abstract":"<div><div>Given a graph <em>G</em>, the parameters <span><math><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and <span><math><mi>ω</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> respectively denote the chromatic number and the clique number of <em>G</em>. A function <span><math><mi>f</mi><mo>:</mo><mi>N</mi><mo>→</mo><mi>N</mi></math></span> such that <span><math><mi>f</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>=</mo><mn>1</mn></math></span> and <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>≥</mo><mi>x</mi></math></span>, for all <span><math><mi>x</mi><mo>∈</mo><mi>N</mi></math></span> is called a <em>χ-binding function</em> for the given class of graphs <span><math><mi>G</mi></math></span> if every <span><math><mi>G</mi><mo>∈</mo><mi>G</mi></math></span> satisfies <span><math><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>f</mi><mo>(</mo><mi>ω</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span>, and the <em>smallest χ-binding function</em> <span><math><msup><mrow><mi>f</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> for <span><math><mi>G</mi></math></span> is defined as <span><math><msup><mrow><mi>f</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>max</mi><mo></mo><mo>{</mo><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mi>G</mi><mo>∈</mo><mi>G</mi><mtext> and </mtext><mi>ω</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>x</mi><mo>}</mo></math></span>. In general, the problem of obtaining the smallest <em>χ</em>-binding function for the given class of graphs seems to be extremely hard, and only a few classes of graphs are studied in this direction. In this paper, we study the class of (<span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>, gem)-free graphs, and prove that the function <span><math><mi>ϕ</mi><mo>:</mo><mi>N</mi><mo>→</mo><mi>N</mi></math></span> defined by <span><math><mi>ϕ</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>=</mo><mn>1</mn></math></span>, <span><math><mi>ϕ</mi><mo>(</mo><mn>2</mn><mo>)</mo><mo>=</mo><mn>4</mn></math></span>, <span><math><mi>ϕ</mi><mo>(</mo><mn>3</mn><mo>)</mo><mo>=</mo><mn>6</mn></math></span> and <span><math><mi>ϕ</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mrow><mo>⌈</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>(</mo><mn>5</mn><mi>x</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>⌉</mo></mrow></math></span>, for <span><math><mi>x</mi><mo>≥</mo><mn>4</mn></math></span> is the smallest <em>χ</em>-binding function for the class of (<span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>, gem)-free graphs. Also we completely characterize the class of (<span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>, gem)-free","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 10","pages":"Article 114554"},"PeriodicalIF":0.7,"publicationDate":"2025-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143900018","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":"Affine groups as flag-transitive and point-primitive automorphism groups of symmetric designs","authors":"Seyed Hassan Alavi , Mohsen Bayat , Ashraf Daneshkhah , Alessandro Montinaro","doi":"10.1016/j.disc.2025.114555","DOIUrl":"10.1016/j.disc.2025.114555","url":null,"abstract":"<div><div>In this article, we investigate symmetric designs admitting flag-transitive and point-primitive affine automorphism groups. We prove that if a flag-transitive automorphism group <em>G</em> of a symmetric <span><math><mo>(</mo><mi>v</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math></span> design with <em>λ</em> prime is point-primitive of affine type, then <span><math><mi>G</mi><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mn>6</mn></mrow></msup><mo>:</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>6</mn></mrow></msub></math></span> and <span><math><mo>(</mo><mi>v</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>λ</mi><mo>)</mo><mo>=</mo><mo>(</mo><mn>16</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, or <em>G</em> is a subgroup of <span><math><mi>A</mi><mi>Γ</mi><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span> for some odd prime power <em>q</em>. In conclusion, we present a classification of flag-transitive and point-primitive symmetric designs with <em>λ</em> prime, which says that such an incidence structure is a projective space <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, it has parameter set <span><math><mo>(</mo><mn>15</mn><mo>,</mo><mn>7</mn><mo>,</mo><mn>3</mn><mo>)</mo></math></span>, <span><math><mo>(</mo><mn>7</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, <span><math><mo>(</mo><mn>11</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, <span><math><mo>(</mo><mn>11</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, <span><math><mo>(</mo><mn>16</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span> or <span><math><mo>(</mo><mn>45</mn><mo>,</mo><mn>12</mn><mo>,</mo><mn>3</mn><mo>)</mo></math></span>, or <span><math><mi>v</mi><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> where <em>p</em> is an odd prime and the automorphism group is a subgroup of <span><math><mi>A</mi><mi>Γ</mi><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 10","pages":"Article 114555"},"PeriodicalIF":0.7,"publicationDate":"2025-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143900019","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":"Nice vertices in cubic graphs","authors":"Wuxian Chen , Fuliang Lu , Heping Zhang","doi":"10.1016/j.disc.2025.114553","DOIUrl":"10.1016/j.disc.2025.114553","url":null,"abstract":"<div><div>A subgraph <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> of a graph <em>G</em> is <em>nice</em> if <span><math><mi>G</mi><mo>−</mo><mi>V</mi><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></math></span> has a perfect matching. Nice subgraphs play a vital role in the theory of ear decomposition and matching minors of matching covered graphs. A vertex <em>u</em> of a cubic graph is <em>nice</em> if <em>u</em> and its neighbors induce a nice subgraph. D. Král et al. (2010) <span><span>[9]</span></span> showed that each vertex of a cubic brick is nice. It is natural to ask how many nice vertices a matching covered cubic graph has. In this paper, using some basic results of matching covered graphs, we prove that if a non-bipartite cubic graph <em>G</em> is 2-connected, then <em>G</em> has at least 4 nice vertices; if <em>G</em> is 3-connected and <span><math><mi>G</mi><mo>≠</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>, then <em>G</em> has at least 6 nice vertices. We also determine all the corresponding extremal graphs. For a cubic bipartite graph <em>G</em> with bipartition <span><math><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo></math></span>, a pair of vertices <span><math><mi>a</mi><mo>∈</mo><mi>A</mi></math></span> and <span><math><mi>b</mi><mo>∈</mo><mi>B</mi></math></span> is called a <em>nice pair</em> if <em>a</em> and <em>b</em> together with their neighbors induce a nice subgraph. We show that a connected cubic bipartite graph <em>G</em> is a brace if and only if each pair of vertices in distinct color classes is a nice pair. In general, we prove that <em>G</em> has at least 9 nice pairs of vertices and <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>3</mn></mrow></msub></math></span> is the only extremal graph.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 10","pages":"Article 114553"},"PeriodicalIF":0.7,"publicationDate":"2025-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143878956","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":"Turán number of complete bipartite graphs with bounded matching number","authors":"Huan Luo, Xiamiao Zhao, Mei Lu","doi":"10.1016/j.disc.2025.114552","DOIUrl":"10.1016/j.disc.2025.114552","url":null,"abstract":"<div><div>Let <span><math><mi>F</mi></math></span> be a family of graphs. A graph <em>G</em> is <span><math><mi>F</mi></math></span>-free if <em>G</em> does not contain any <span><math><mi>F</mi><mo>∈</mo><mi>F</mi></math></span> as a subgraph. The Turán number <span><math><mtext>ex</mtext><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> is the maximum number of edges in an <em>n</em>-vertex <span><math><mi>F</mi></math></span>-free graph. Let <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span> be the matching consisting of <em>s</em> independent edges. Recently, Alon and Frankl determined the exact value of <span><math><mtext>ex</mtext><mo>(</mo><mi>n</mi><mo>,</mo><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>}</mo><mo>)</mo></math></span>. Gerbner obtained several results about <span><math><mtext>ex</mtext><mo>(</mo><mi>n</mi><mo>,</mo><mo>{</mo><mi>F</mi><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>}</mo><mo>)</mo></math></span> when <em>F</em> satisfies certain properties. In this paper, we determine the exact value of <span><math><mtext>ex</mtext><mo>(</mo><mi>n</mi><mo>,</mo><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>ℓ</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>}</mo><mo>)</mo></math></span> when <span><math><mi>s</mi><mo>,</mo><mi>n</mi></math></span> are large enough for every <span><math><mn>3</mn><mo>≤</mo><mi>ℓ</mi><mo>≤</mo><mi>t</mi></math></span>. When <em>n</em> is large enough, we also show that <span><math><mtext>ex</mtext><mo>(</mo><mi>n</mi><mo>,</mo><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>}</mo><mo>)</mo><mo>=</mo><mi>n</mi><mo>+</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>s</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow><mo>−</mo><mrow><mo>⌈</mo><mfrac><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></mrow></math></span> for <span><math><mi>s</mi><mo>≥</mo><mn>12</mn></math></span> and <span><math><mtext>ex</mtext><mo>(</mo><mi>n</mi><mo>,</mo><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>t</mi></mrow></msub><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>}</mo><mo>)</mo><mo>=</mo><mi>n</mi><mo>+</mo><mo>(</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo>)</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>s</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow><mo>−</mo><mrow><mo>⌈</mo><mfrac><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></mrow></math></span> when <span><math><mi>t</mi><mo>≥</mo><mn>3</mn></math></span> and <em>s</em> is large enough.</div><","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 9","pages":"Article 114552"},"PeriodicalIF":0.7,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143867927","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 the roots of maximal matching polynomials","authors":"Matt Burnham, Aysel Erey","doi":"10.1016/j.disc.2025.114547","DOIUrl":"10.1016/j.disc.2025.114547","url":null,"abstract":"<div><div>A <em>maximal matching</em> of a graph <em>G</em> is a matching of <em>G</em> which is not properly contained in any other matching of <em>G</em>. Let <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the number of maximal matchings of size <em>k</em> in <em>G</em>. The <em>maximal matching polynomial</em> of <em>G</em> is defined by <span><math><mi>m</mi><mo>(</mo><mi>G</mi><mo>;</mo><mi>x</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi>m</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span>. It is known that maximal matching polynomials generalize the well-known matching polynomials, as the matching polynomial of every graph can be obtained from the maximal matching polynomial of some other graph. While the roots of matching polynomials have been extensively studied and well understood, the study of the roots of maximal matching polynomials has not been developed. In this article, we study the location of the roots of these polynomials. We show that maximal matching polynomials of paths and cycles have only real roots, and provide interlacing relations for their roots. On the other hand, unlike matching polynomials, maximal matching polynomials can have non-real roots, and we provide an infinite family of graphs whose maximal matching polynomials have non-real roots.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 9","pages":"Article 114547"},"PeriodicalIF":0.7,"publicationDate":"2025-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143863561","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":"Descents in powers of permutations","authors":"Kassie Archer, Aaron Geary","doi":"10.1016/j.disc.2025.114551","DOIUrl":"10.1016/j.disc.2025.114551","url":null,"abstract":"<div><div>We consider a few special cases of the more general question: How many permutations <span><math><mi>π</mi><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> have the property that <span><math><msup><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> has <em>j</em> descents for some <em>j</em>? In this paper, we first enumerate Grassmannian permutations <em>π</em> by the number of descents in <span><math><msup><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. We then consider all permutations whose square has exactly one descent, fully enumerating when the descent is “small” and providing a lower bound in the general case. Finally, we enumerate permutations whose square or cube has the maximum number of descents, and finish the paper with a few future directions for study.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 9","pages":"Article 114551"},"PeriodicalIF":0.7,"publicationDate":"2025-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143863312","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":"Intersecting families with covering number five","authors":"Peter Frankl , Jian Wang","doi":"10.1016/j.disc.2025.114546","DOIUrl":"10.1016/j.disc.2025.114546","url":null,"abstract":"<div><div>A family <span><math><mi>F</mi><mo>⊂</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow></math></span> is called intersecting if any two members of it have non-empty intersection. The covering number of <span><math><mi>F</mi></math></span> is defined as the minimum integer <em>p</em> such that there exists <span><math><mi>T</mi><mo>⊂</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span> satisfying <span><math><mo>|</mo><mi>T</mi><mo>|</mo><mo>=</mo><mi>p</mi></math></span> and <span><math><mi>T</mi><mo>∩</mo><mi>F</mi><mo>≠</mo><mo>∅</mo></math></span> for all <span><math><mi>F</mi><mo>∈</mo><mi>F</mi></math></span>. Define <span><math><mi>m</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span> as the maximum size of an intersecting family <span><math><mi>F</mi><mo>⊂</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow></math></span> with covering number at least <em>p</em>. The value of <span><math><mi>m</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span> is only known for <span><math><mi>p</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn></math></span>. About thirty years ago, <span><math><mi>m</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mn>5</mn><mo>)</mo></math></span> was determined asymptotically by the first author, Ota and Tokushige. In the present paper, we determine <span><math><mi>m</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mn>5</mn><mo>)</mo></math></span> for <span><math><mi>k</mi><mo>≥</mo><mn>69</mn></math></span> and <span><math><mi>n</mi><mo>≥</mo><mn>5</mn><msup><mrow><mi>k</mi></mrow><mrow><mn>6</mn></mrow></msup></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 9","pages":"Article 114546"},"PeriodicalIF":0.7,"publicationDate":"2025-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143863311","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":"Group action approaches in Erdős quotient set problem","authors":"Will Burstein","doi":"10.1016/j.disc.2025.114505","DOIUrl":"10.1016/j.disc.2025.114505","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> denote the finite field of <em>q</em> elements. Let <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>d</mi></mrow></msubsup></math></span> be the <em>d</em>-dimensional vector space over the field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. Let <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>:</mo><mo>=</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>∖</mo><mo>{</mo><mn>0</mn><mo>}</mo></math></span>. Let <span><math><msup><mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>:</mo><mo>=</mo><mo>{</mo><msup><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>:</mo><mi>t</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>}</mo></math></span>. For <span><math><mi>E</mi><mo>⊂</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>d</mi></mrow></msubsup></math></span>, denote the distance set by <span><math><mi>Δ</mi><mo>(</mo><mi>E</mi><mo>)</mo><mo>:</mo><mo>=</mo><mo>{</mo><mo>‖</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo>‖</mo><mo>:</mo><mo>=</mo><msup><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>−</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mo>⋯</mo><mo>+</mo><msup><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>−</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>:</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>E</mi><mo>}</mo></math></span>. Denote the Erdős quotient set by <span><math><mfrac><mrow><mi>Δ</mi><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mrow><mi>Δ</mi><mo>(</mo><mi>E</mi><mo>)</mo></mrow></mfrac><mo>:</mo><mo>=</mo><mo>{</mo><mfrac><mrow><mi>s</mi></mrow><mrow><mi>t</mi></mrow></mfrac><mo>:</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi>Δ</mi><mo>(</mo><mi>E</mi><mo>)</mo><mo>,</mo><mi>t</mi><mo>≠</mo><mn>0</mn><mo>}</mo><mo>=</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>.</div><div>The Erdős quotient set problem was introduced in <span><span>[13]</span></span> where it was shown that for even <span><math><mi>d</mi><mo>≥</mo><mn>2</mn></math></span>, if <span><math><mi>E</mi><mo>⊂</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msubsup></math></span> such that <span><math><mo>|</mo><mi>E</mi><mo>|</mo><mo>≫</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>d</mi><mo>/</mo><mn>2</mn></mrow></msup></math></span>, then <span><math><mfrac><mrow><mi>Δ</mi><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mrow><mi>Δ</mi><mo>(</mo><mi>E</mi><mo>)</mo></mrow></mfrac><mo>=</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 9","pages":"Article 114505"},"PeriodicalIF":0.7,"publicationDate":"2025-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143863291","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":"Bounds for asymmetric nested orthogonal arrays","authors":"Xiao Lin, Shanqi Pang, Guangzhou Chen","doi":"10.1016/j.disc.2025.114549","DOIUrl":"10.1016/j.disc.2025.114549","url":null,"abstract":"<div><div>Nested orthogonal arrays (NOAs) are more and more widely used in diverse experiments. An important problem in the study of NOAs is to determine the minimal number of runs, i.e., to find the bounds on the rows for NOAs. These bounds are quite powerful in proving nonexistence. Although the bounds for symmetric NOAs were derived over a decade, the bounds for asymmetric NOAs remain an open problem. This article presents the bounds for asymmetric NOAs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 9","pages":"Article 114549"},"PeriodicalIF":0.7,"publicationDate":"2025-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143859144","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}
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