Journal of Combinatorial Theory Series A最新文献

Dominance complexes, neighborhood complexes and combinatorial Alexander duals 支配复合体、邻域复合体和组合亚历山大对偶

IF 0.9 2区数学

Journal of Combinatorial Theory Series A Pub Date : 2024-11-16 DOI: 10.1016/j.jcta.2024.105978

Takahiro Matsushita , Shun Wakatsuki

{"title":"Dominance complexes, neighborhood complexes and combinatorial Alexander duals","authors":"Takahiro Matsushita , Shun Wakatsuki","doi":"10.1016/j.jcta.2024.105978","DOIUrl":"10.1016/j.jcta.2024.105978","url":null,"abstract":"<div><div>We show that the dominance complex <math><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo></math> of a graph G coincides with the combinatorial Alexander dual of the neighborhood complex <math><mi>N</mi><mo>(</mo><mover><mrow><mi>G</mi></mrow><mo>‾</mo></mover><mo>)</mo></math> of the complement of G. Using this, we obtain a relation between the chromatic number <math><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math> of G and the homology group of <math><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo></math>. We also obtain several known results related to dominance complexes from well-known facts of neighborhood complexes. After that, we suggest a new method for computing the homology groups of the dominance complexes, using independence complexes of simple graphs. We show that several known computations of homology groups of dominance complexes can be reduced to known computations of independence complexes. Finally, we determine the homology group of <math><mi>D</mi><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>×</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></math> by determining the homotopy types of the independence complex of <math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>×</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>×</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub></math>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"211 ","pages":"Article 105978"},"PeriodicalIF":0.9,"publicationDate":"2024-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652948","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Upper bounds for the number of substructures in finite geometries from the container method 从容器法看有限几何中子结构数量的上界

IF 0.9 2区数学

Journal of Combinatorial Theory Series A Pub Date : 2024-11-06 DOI: 10.1016/j.jcta.2024.105968

Sam Mattheus, Geertrui Van de Voorde

{"title":"Upper bounds for the number of substructures in finite geometries from the container method","authors":"Sam Mattheus, Geertrui Van de Voorde","doi":"10.1016/j.jcta.2024.105968","DOIUrl":"10.1016/j.jcta.2024.105968","url":null,"abstract":"<div><div>We use techniques from algebraic and extremal combinatorics to derive upper bounds on the number of independent sets in several (hyper)graphs arising from finite geometry. In this way, we obtain asymptotically sharp upper bounds for partial ovoids and EKR-sets of flags in polar spaces, line spreads in <math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>2</mn><mi>r</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>q</mi><mo>)</mo></math> and plane spreads in <math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>5</mn><mo>,</mo><mi>q</mi><mo>)</mo></math>, and caps in <math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>)</mo></math>. The latter result extends work due to Roche-Newton and Warren [21] and Bhowmick and Roche-Newton [6].</div><div>Finally, we investigate caps in p-random subsets of <math><mrow><mi>PG</mi></mrow><mo>(</mo><mi>r</mi><mo>,</mo><mi>q</mi><mo>)</mo></math>, which parallels recent work for arcs in projective planes by Bhowmick and Roche-Newton, and Roche-Newton and Warren [6], [21], and arcs in projective spaces by Chen, Liu, Nie and Zeng [8].</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"210 ","pages":"Article 105968"},"PeriodicalIF":0.9,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142593647","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

The vector space generated by permutations of a trade or a design 由交易或设计的排列组合产生的向量空间

IF 0.9 2区数学

Journal of Combinatorial Theory Series A Pub Date : 2024-11-06 DOI: 10.1016/j.jcta.2024.105969

E. Ghorbani , S. Kamali , G.B. Khosrovshahi

{"title":"The vector space generated by permutations of a trade or a design","authors":"E. Ghorbani , S. Kamali , G.B. Khosrovshahi","doi":"10.1016/j.jcta.2024.105969","DOIUrl":"10.1016/j.jcta.2024.105969","url":null,"abstract":"<div><div>Motivated by a classical result of Graver and Jurkat (1973) and Graham, Li, and Li (1980) in combinatorial design theory, which states that the permutations of t-<math><mo>(</mo><mi>v</mi><mo>,</mo><mi>k</mi><mo>)</mo></math> minimal trades generate the vector space of all t-<math><mo>(</mo><mi>v</mi><mo>,</mo><mi>k</mi><mo>)</mo></math> trades, we investigate the vector space spanned by permutations of an arbitrary trade. We prove that this vector space possesses a decomposition as a direct sum of subspaces formed in the same way by a specific family of so-called total trades. As an application, we demonstrate that for any t-<math><mo>(</mo><mi>v</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math> design, its permutations can span the vector space generated by all t-<math><mo>(</mo><mi>v</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math> designs for sufficiently large values of v. In other words, any t-<math><mo>(</mo><mi>v</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math> design, or even any t-trade, can be expressed as a linear combination of permutations of a fixed t-design. This substantially extends a result by Ghodrati (2019), who proved the same result for Steiner designs.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"210 ","pages":"Article 105969"},"PeriodicalIF":0.9,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142593648","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Some conjectures of Ballantine and Merca on truncated sums and the minimal excludant in congruences classes 巴兰廷和梅尔卡关于截断和以及全等类中最小排除因子的一些猜想

IF 0.9 2区数学

Journal of Combinatorial Theory Series A Pub Date : 2024-11-05 DOI: 10.1016/j.jcta.2024.105967

Olivia X.M. Yao

引用次数: 0

Reconstruction of hypermatrices from subhypermatrices 从次皮质重建高皮质

IF 0.9 2区数学

Journal of Combinatorial Theory Series A Pub Date : 2024-10-22 DOI: 10.1016/j.jcta.2024.105966

Wenjie Zhong , Xiande Zhang

{"title":"Reconstruction of hypermatrices from subhypermatrices","authors":"Wenjie Zhong , Xiande Zhang","doi":"10.1016/j.jcta.2024.105966","DOIUrl":"10.1016/j.jcta.2024.105966","url":null,"abstract":"<div><div>For a given n, what is the smallest number k such that every sequence of length n is determined by the multiset of all its k-subsequences? This is called the k-deck problem for sequence reconstruction, and has been generalized to the two-dimensional case – reconstruction of <math><mi>n</mi><mo>×</mo><mi>n</mi></math>-matrices from submatrices. Previous works show that the smallest k is at most <math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>)</mo></math> for sequences and at most <math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup><mo>)</mo></math> for matrices. We study this k-deck problem for general dimension d and prove that, the smallest k is at most <math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mi>d</mi></mrow><mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></msup><mo>)</mo></math> for reconstructing any d dimensional hypermatrix of order n from the multiset of all its subhypermatrices of order k.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"209 ","pages":"Article 105966"},"PeriodicalIF":0.9,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530619","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Direct constructions of column-orthogonal strong orthogonal arrays 列正交强正交阵列的直接构造

IF 0.9 2区数学

Journal of Combinatorial Theory Series A Pub Date : 2024-10-18 DOI: 10.1016/j.jcta.2024.105965

Jingjun Bao , Lijun Ji , Juanjuan Xu

引用次数: 0

Indecomposable combinatorial games 不可分解的组合博弈

IF 0.9 2区数学

Journal of Combinatorial Theory Series A Pub Date : 2024-10-15 DOI: 10.1016/j.jcta.2024.105964

Michael Fisher , Neil A. McKay , Rebecca Milley , Richard J. Nowakowski , Carlos P. Santos

引用次数: 0

Point-line geometries related to binary equidistant codes 与二进制等距码有关的点线几何图形

IF 0.9 2区数学

Journal of Combinatorial Theory Series A Pub Date : 2024-10-04 DOI: 10.1016/j.jcta.2024.105962

Mark Pankov , Krzysztof Petelczyc , Mariusz Żynel

引用次数: 0

Neighborly partitions, hypergraphs and Gordon's identities 邻接分区、超图和戈登等式

IF 0.9 2区数学

Journal of Combinatorial Theory Series A Pub Date : 2024-10-04 DOI: 10.1016/j.jcta.2024.105963

Pooneh Afsharijoo , Hussein Mourtada

引用次数: 0

On locally n × n grid graphs 在局部 n×n 网格图上

IF 0.9 2区数学

Journal of Combinatorial Theory Series A Pub Date : 2024-09-26 DOI: 10.1016/j.jcta.2024.105957

Carmen Amarra , Wei Jin , Cheryl E. Praeger

{"title":"On locally n × n grid graphs","authors":"Carmen Amarra , Wei Jin , Cheryl E. Praeger","doi":"10.1016/j.jcta.2024.105957","DOIUrl":"10.1016/j.jcta.2024.105957","url":null,"abstract":"<div><div>We investigate locally <math><mi>n</mi><mo>×</mo><mi>n</mi></math> grid graphs, that is, graphs in which the neighbourhood of any vertex is the Cartesian product of two complete graphs on n vertices. We consider the subclass of these graphs for which each pair of vertices at distance two is joined by sufficiently many paths of length 2. The number of such paths is known to be at most 2n by previous work of Blokhuis and Brouwer. We show that if each pair is joined by at least <math><mn>2</mn><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math> such paths then the diameter is at most 3 and we give a tight upper bound on the order of the graphs. We show that graphs meeting this upper bound are distance-regular antipodal covers of complete graphs. We exhibit an infinite family of such graphs which are locally <math><mi>n</mi><mo>×</mo><mi>n</mi></math> grid for odd prime powers n, and apply these results to locally <math><mn>5</mn><mo>×</mo><mn>5</mn></math> grid graphs to obtain a classification for the case where either all μ-graphs (induced subgraphs on the set of common neighbours of two vertices at distance two) have order at least 8 or all μ-graphs have order c for some constant c.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"209 ","pages":"Article 105957"},"PeriodicalIF":0.9,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142322889","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0