Combinatorica最新文献_第2页

Two Disjoint Alternating Paths in Bipartite Graphs: Conformal Crosses 二部图中两条不相交的交变路径：共形交叉

IF 1.1 2区数学

Combinatorica Pub Date : 2026-05-05 DOI: 10.1007/s00493-026-00211-4

Archontia C. Giannopoulou, Sebastian Wiederrecht

引用次数: 0

Tangle-tree duality in infinite graphs 无穷图中的缠结树对偶性

IF 1.1 2区数学

Combinatorica Pub Date : 2026-05-05 DOI: 10.1007/s00493-026-00210-5

Sandra Albrechtsen

引用次数: 0

Exact Results on Traces of Sets 集迹的精确结果

IF 1.1 2区数学

Combinatorica Pub Date : 2026-04-27 DOI: 10.1007/s00493-026-00209-y

Mingze Li, Jie Ma, Mingyuan Rong

引用次数: 0

Explosive appearance of cores and bootstrap percolation on lattices 岩心的爆炸外观和晶格上的自举渗流

IF 1.1 2区数学

Combinatorica Pub Date : 2026-04-27 DOI: 10.1007/s00493-026-00213-2

Ivailo Hartarsky, Lyuben Lichev

{"title":"Explosive appearance of cores and bootstrap percolation on lattices","authors":"Ivailo Hartarsky, Lyuben Lichev","doi":"10.1007/s00493-026-00213-2","DOIUrl":"https://doi.org/10.1007/s00493-026-00213-2","url":null,"abstract":"Consider the process where the n vertices of a square 2-dimensional torus appear consecutively in a random order. We show that typically the size of the 3-core of the corresponding induced unit-distance graph transitions from 0 to $$n-o(n)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mi>o</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> within a single step. Equivalently, by infecting the vertices of the torus in a random order, under two-neighbour bootstrap percolation, the size of the infected set transitions instantaneously from o ( n ) to n . This hitting time result answers a question of Benjamini. We also study the much more challenging and general setting of bootstrap percolation on two-dimensional lattices for a variety of finite-range infection rules. In this case, powerful but fragile bootstrap percolation tools such as the rectangles process and the Aizenman–Lebowitz lemma become unavailable. We develop a new method complementing and replacing these standard techniques, thus allowing us to prove the above hitting time result for a wide family of threshold bootstrap percolation rules on the 2-dimensional square lattice, including neighbourhoods given by large $$ell ^p$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> balls for $$pin [1,infty ]$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>[</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> .","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"151 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147751500","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

Explicit Constructions of Optimal Blocking Sets and Minimal Codes 最优块集和最小码的显式构造

IF 1.1 2区数学

Combinatorica Pub Date : 2026-03-12 DOI: 10.1007/s00493-026-00202-5

Anurag Bishnoi, István Tomon

{"title":"Explicit Constructions of Optimal Blocking Sets and Minimal Codes","authors":"Anurag Bishnoi, István Tomon","doi":"10.1007/s00493-026-00202-5","DOIUrl":"https://doi.org/10.1007/s00493-026-00202-5","url":null,"abstract":"A strong s -blocking set in a projective space is a set of points that intersects each codimension- s subspace in a spanning set of the subspace. We present an explicit construction of such sets in a $$(k - 1)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -dimensional projective space over $$mathbb {F}_q$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>q</mml:mi> </mml:msub> </mml:math> of size $$O_s(q^s k)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>q</mml:mi> <mml:mi>s</mml:mi> </mml:msup> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , which is optimal up to the constant factor depending on s . This also yields an optimal explicit construction of affine blocking sets in $$mathbb {F}_q^k$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msubsup> <mml:mi>F</mml:mi> <mml:mi>q</mml:mi> <mml:mi>k</mml:mi> </mml:msubsup> </mml:math> with respect to codimension- $$(s+1)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> affine subspaces, and of s -minimal codes. Our approach is motivated by a recent construction of Alon, Bishnoi, Das, and Neri of strong 1-blocking sets, which uses expander graphs with a carefully chosen set of vectors as their vertex set. The main novelty of our work lies in constructing specific hypergraphs on top of these expander graphs, where tree-like configurations correspond to strong s -blocking sets. We also discuss some connections to size-Ramsey numbers of hypergraphs, which might be of independent interest.","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"266 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147461906","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

Modified Macdonald Polynomials and $$mu $$-Mahonian Statistics 修正麦克唐纳多项式和$$mu $$ -马奥尼统计

IF 1.1 2区数学

Combinatorica Pub Date : 2026-03-12 DOI: 10.1007/s00493-026-00204-3

Emma Yu Jin, Xiaowei Lin

引用次数: 0

Leaf-to-leaf paths and cycles in degree-critical graphs 度临界图中的叶到叶路径和循环

IF 1.1 2区数学

Combinatorica Pub Date : 2026-03-04 DOI: 10.1007/s00493-026-00205-2

Francesco Di Braccio, Kyriakos Katsamaktsis, Jie Ma, Alexandru Malekshahian, Ziyuan Zhao

{"title":"Leaf-to-leaf paths and cycles in degree-critical graphs","authors":"Francesco Di Braccio, Kyriakos Katsamaktsis, Jie Ma, Alexandru Malekshahian, Ziyuan Zhao","doi":"10.1007/s00493-026-00205-2","DOIUrl":"https://doi.org/10.1007/s00493-026-00205-2","url":null,"abstract":"An <jats:italic>n</jats:italic> -vertex graph is <jats:italic>degree 3-critical</jats:italic> if it has <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$2n - 2$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> edges and no proper induced subgraph with minimum degree at least 3. In 1988, Erdős, Faudree, Gyárfás, and Schelp asked whether one can always find cycles of all short lengths in these graphs, which was disproven by Narins, Pokrovskiy, and Szabó through a construction based on leaf-to-leaf paths in trees whose vertices have degree either 1 or 3. They went on to suggest several weaker conjectures about cycle lengths in degree 3-critical graphs and leaf-to-leaf path lengths in these so-called 1-3 trees. We resolve three of their questions either fully or up to a constant factor. Our main results are the following: <jats:list list-type=\"bullet\"> <jats:list-item> every <jats:italic>n</jats:italic> -vertex degree 3-critical graph has <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$Omega (log n)$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>(</mml:mo> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> distinct cycle lengths; </jats:list-item> <jats:list-item> every tree with maximum degree <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$Delta ge 3$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$ell $$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>ℓ</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> leaves has at least <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$log _{Delta -1}, ((Delta -2)ell )$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msub> <mml:mo>log</mml:mo> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mspace/> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>ℓ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> distinct leaf-to-leaf path lengths; </jats:list-item> <jats:list-item> for every integer <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$Nge 1$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> </jats:alternative","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"55 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147359909","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

Orientations of Cycles in Digraphs of High Chromatic Number and High Minimum Out-Degree 高色数和高最小出度有向图中圈的取向

IF 1.1 2区数学

Combinatorica Pub Date : 2026-03-04 DOI: 10.1007/s00493-026-00207-0

Hidde Koerts, Benjamin Moore, Sophie Spirkl

引用次数: 0

Erdős-Pósa property of A-paths in unoriented group-labelled graphs Erdős-Pósa无向群标记图中a路径的性质

IF 1.1 2区数学

Combinatorica Pub Date : 2026-03-04 DOI: 10.1007/s00493-026-00203-4

O-joung Kwon, Youngho Yoo

引用次数: 0

Small Families of Partially Shattering Permutations 部分破碎排列的小家族