Journal of Combinatorial Theory Series B最新文献_第2页

Toward a density Corrádi–Hajnal theorem for degenerate hypergraphs

IF 1.2 1区数学

Journal of Combinatorial Theory Series B Pub Date : 2025-01-23 DOI: 10.1016/j.jctb.2025.01.001

Jianfeng Hou , Caiyun Hu , Heng Li , Xizhi Liu , Caihong Yang , Yixiao Zhang

{"title":"Toward a density Corrádi–Hajnal theorem for degenerate hypergraphs","authors":"Jianfeng Hou , Caiyun Hu , Heng Li , Xizhi Liu , Caihong Yang , Yixiao Zhang","doi":"10.1016/j.jctb.2025.01.001","DOIUrl":"10.1016/j.jctb.2025.01.001","url":null,"abstract":"<div><div>Given an r-graph F with <math><mi>r</mi><mo>≥</mo><mn>2</mn></math>, let <math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mo>(</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo><mi>F</mi><mo>)</mo></math> denote the maximum number of edges in an n-vertex r-graph with at most t pairwise vertex-disjoint copies of F. Extending several old results and complementing prior work [34] on nondegenerate hypergraphs, we initiate a systematic study on <math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mo>(</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo><mi>F</mi><mo>)</mo></math> for degenerate hypergraphs F.</div><div>For a broad class of degenerate hypergraphs F, we present near-optimal upper bounds for <math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mo>(</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo><mi>F</mi><mo>)</mo></math> when n is sufficiently large and t lies in intervals <math><mo>[</mo><mn>0</mn><mo>,</mo><mfrac><mrow><mi>ε</mi><mo>⋅</mo><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfrac><mo>]</mo></math>, <math><mo>[</mo><mfrac><mrow><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></mrow><mrow><mi>ε</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfrac><mo>,</mo><mi>ε</mi><mi>n</mi><mo>]</mo></math>, and <math><mo>[</mo><mo>(</mo><mn>1</mn><mo>−</mo><mi>ε</mi><mo>)</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mi>v</mi><mo>(</mo><mi>F</mi><mo>)</mo></mrow></mfrac><mo>,</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mi>v</mi><mo>(</mo><mi>F</mi><mo>)</mo></mrow></mfrac><mo>]</mo></math>, where <math><mi>ε</mi><mo>></mo><mn>0</mn></math> is a constant depending only on F. Our results reveal very different structures for extremal constructions across the three intervals, and we provide characterizations of extremal constructions within the first interval. Additionally, we characterize extremal constructions within the second interval for graphs. Our proof for the first interval also applies to a special class of nondegenerate hypergraphs, including those with undetermined Turán densities, partially improving a result in [34].</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"172 ","pages":"Pages 221-262"},"PeriodicalIF":1.2,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143092870","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

The next case of Andrásfai's conjecture

IF 1.2 1区数学

Journal of Combinatorial Theory Series B Pub Date : 2025-01-14 DOI: 10.1016/j.jctb.2024.12.010

Tomasz Łuczak , Joanna Polcyn , Christian Reiher

引用次数: 0

Weak saturation in graphs: A combinatorial approach

IF 1.2 1区数学

Journal of Combinatorial Theory Series B Pub Date : 2025-01-13 DOI: 10.1016/j.jctb.2024.12.007

Nikolai Terekhov , Maksim Zhukovskii

{"title":"Weak saturation in graphs: A combinatorial approach","authors":"Nikolai Terekhov , Maksim Zhukovskii","doi":"10.1016/j.jctb.2024.12.007","DOIUrl":"10.1016/j.jctb.2024.12.007","url":null,"abstract":"<div><div>The weak saturation number <math><mrow><mi>wsat</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math> is the minimum number of edges in a graph on n vertices such that all the missing edges can be activated sequentially so that each new edge creates a copy of F. In contrast to previous algebraic approaches, we present a new combinatorial approach to prove lower bounds for weak saturation numbers that allows to establish worst-case tight (up to constant additive terms) general lower bounds as well as to get exact values of the weak saturation numbers for certain graph families. It is known (Alon, 1985) that, for every F, there exists <math><msub><mrow><mi>c</mi></mrow><mrow><mi>F</mi></mrow></msub></math> such that <math><mrow><mi>wsat</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>F</mi></mrow></msub><mi>n</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo></math>. Our lower bounds imply that all values in the interval <math><mo>[</mo><mfrac><mrow><mi>δ</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>δ</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>,</mo><mi>δ</mi><mo>−</mo><mn>1</mn><mo>]</mo></math> with step size <math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>δ</mi><mo>+</mo><mn>1</mn></mrow></mfrac></math> are achievable by <math><msub><mrow><mi>c</mi></mrow><mrow><mi>F</mi></mrow></msub></math> for graphs F with minimum degree δ (while any value outside this interval is not achievable).</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"172 ","pages":"Pages 146-167"},"PeriodicalIF":1.2,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143127971","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Kővári-Sós-Turán theorem for hereditary families

IF 1.2 1区数学

Journal of Combinatorial Theory Series B Pub Date : 2025-01-13 DOI: 10.1016/j.jctb.2024.12.009

Zach Hunter , Aleksa Milojević , Benny Sudakov , István Tomon

{"title":"Kővári-Sós-Turán theorem for hereditary families","authors":"Zach Hunter , Aleksa Milojević , Benny Sudakov , István Tomon","doi":"10.1016/j.jctb.2024.12.009","DOIUrl":"10.1016/j.jctb.2024.12.009","url":null,"abstract":"<div><div>The celebrated Kővári-Sós-Turán theorem states that any n-vertex graph containing no copy of the complete bipartite graph <math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>s</mi></mrow></msub></math> has at most <math><msub><mrow><mi>O</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn><mo>−</mo><mn>1</mn><mo>/</mo><mi>s</mi></mrow></msup><mo>)</mo></math> edges. In the past two decades, motivated by the applications in discrete geometry and structural graph theory, a number of results demonstrated that this bound can be greatly improved if the graph satisfies certain structural restrictions. We propose the systematic study of this phenomenon, and state the conjecture that if H is a bipartite graph, then an induced H-free and <math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>s</mi></mrow></msub></math>-free graph cannot have much more edges than an H-free graph. We provide evidence for this conjecture by considering trees, cycles, the cube graph, and bipartite graphs with degrees bounded by k on one side, obtaining in all the cases similar bounds as in the non-induced setting. Our results also have applications to the Erdős-Hajnal conjecture, the problem of finding induced <math><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub></math>-free subgraphs with large degree and bounding the average degree of <math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>s</mi></mrow></msub></math>-free graphs which do not contain induced subdivisions of a fixed graph.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"172 ","pages":"Pages 168-197"},"PeriodicalIF":1.2,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143092868","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A half-integral Erdős-Pósa theorem for directed odd cycles

IF 1.2 1区数学

Journal of Combinatorial Theory Series B Pub Date : 2025-01-07 DOI: 10.1016/j.jctb.2024.12.008

Ken-ichi Kawarabayashi , Stephan Kreutzer , O-joung Kwon , Qiqin Xie

引用次数: 0

On the automorphism group of a distance-regular graph

IF 1.2 1区数学

Journal of Combinatorial Theory Series B Pub Date : 2024-12-31 DOI: 10.1016/j.jctb.2024.12.005

László Pyber , Saveliy V. Skresanov

{"title":"On the automorphism group of a distance-regular graph","authors":"László Pyber , Saveliy V. Skresanov","doi":"10.1016/j.jctb.2024.12.005","DOIUrl":"10.1016/j.jctb.2024.12.005","url":null,"abstract":"<div><div>The motion of a graph is the minimal degree of its full automorphism group. Babai conjectured that the motion of a primitive distance-regular graph on n vertices of diameter greater than two is at least <math><mi>n</mi><mo>/</mo><mi>C</mi></math> for some universal constant <math><mi>C</mi><mo>></mo><mn>0</mn></math>, unless the graph is a Johnson or Hamming graph. We prove that the motion of a distance-regular graph of diameter <math><mi>d</mi><mo>≥</mo><mn>3</mn></math> on n vertices is at least <math><mi>C</mi><mi>n</mi><mo>/</mo><msup><mrow><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></mrow><mrow><mn>6</mn></mrow></msup></math> for some universal constant <math><mi>C</mi><mo>></mo><mn>0</mn></math>, unless it is a Johnson, Hamming or crown graph. To show this, we improve an earlier result by Kivva who gave a lower bound on motion of the form <math><mi>n</mi><mo>/</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>d</mi></mrow></msub></math>, where <math><msub><mrow><mi>c</mi></mrow><mrow><mi>d</mi></mrow></msub></math> depends exponentially on d. As a corollary we derive a quasipolynomial upper bound for the size of the automorphism group of a primitive distance-regular graph acting edge-transitively on the graph and on its distance-2 graph. The proofs use elementary combinatorial arguments and do not depend on the classification of finite simple groups.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"172 ","pages":"Pages 94-114"},"PeriodicalIF":1.2,"publicationDate":"2024-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143092866","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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

Aharoni's rainbow cycle conjecture holds up to an additive constant

IF 1.2 1区数学

Journal of Combinatorial Theory Series B Pub Date : 2024-12-30 DOI: 10.1016/j.jctb.2024.12.004

Patrick Hompe, Tony Huynh

{"title":"Aharoni's rainbow cycle conjecture holds up to an additive constant","authors":"Patrick Hompe, Tony Huynh","doi":"10.1016/j.jctb.2024.12.004","DOIUrl":"10.1016/j.jctb.2024.12.004","url":null,"abstract":"<div><div>In 2017, Aharoni proposed the following generalization of the Caccetta-Häggkvist conjecture: if G is a simple n-vertex edge-colored graph with n color classes of size at least r, then G contains a rainbow cycle of length at most <math><mo>⌈</mo><mi>n</mi><mo>/</mo><mi>r</mi><mo>⌉</mo></math>.</div><div>In this paper, we prove that, for fixed r, Aharoni's conjecture holds up to an additive constant. Specifically, we show that for each fixed <math><mi>r</mi><mo>⩾</mo><mn>1</mn></math>, there exists a constant <math><msub><mrow><mi>α</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>∈</mo><mi>O</mi><mo>(</mo><msup><mrow><mi>r</mi></mrow><mrow><mn>5</mn></mrow></msup><msup><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>⁡</mo><mi>r</mi><mo>)</mo></math> such that if G is a simple n-vertex edge-colored graph with n color classes of size at least r, then G contains a rainbow cycle of length at most <math><mi>n</mi><mo>/</mo><mi>r</mi><mo>+</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>r</mi></mrow></msub></math>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"172 ","pages":"Pages 80-93"},"PeriodicalIF":1.2,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143092865","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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

Slow graph bootstrap percolation II: Accelerating properties

IF 1.2 1区数学

Journal of Combinatorial Theory Series B Pub Date : 2024-12-27 DOI: 10.1016/j.jctb.2024.12.006

David Fabian , Patrick Morris , Tibor Szabó

{"title":"Slow graph bootstrap percolation II: Accelerating properties","authors":"David Fabian , Patrick Morris , Tibor Szabó","doi":"10.1016/j.jctb.2024.12.006","DOIUrl":"10.1016/j.jctb.2024.12.006","url":null,"abstract":"<div><div>For a graph H and an n-vertex graph G, the H-bootstrap process on G is the process which starts with G and, at every time step, adds any missing edges on the vertices of G that complete a copy of H. This process eventually stabilises and we are interested in the extremal question raised by Bollobás of determining the maximum running time (number of time steps before stabilising) of this process over all possible choices of n-vertex graph G. In this paper, we initiate a systematic study of the asymptotics of this parameter, denoted <math><msub><mrow><mi>M</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math>, and its dependence on properties of the graph H. Our focus is on H which define relatively fast bootstrap processes, that is, with <math><msub><mrow><mi>M</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math> being at most linear in n. We study the graph class of trees, showing that one can bound <math><msub><mrow><mi>M</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math> by a quadratic function in <math><mi>v</mi><mo>(</mo><mi>T</mi><mo>)</mo></math> for all trees T and all n. We then go on to explore the relationship between the running time of the H-process and the minimum vertex degree and connectivity of H.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"172 ","pages":"Pages 44-79"},"PeriodicalIF":1.2,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143127972","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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

Unexpected automorphisms in direct product graphs 直积图中的意外自同构

IF 1.2 1区数学

Journal of Combinatorial Theory Series B Pub Date : 2024-12-19 DOI: 10.1016/j.jctb.2024.12.003

Yunsong Gan , Weijun Liu , Binzhou Xia

{"title":"Unexpected automorphisms in direct product graphs","authors":"Yunsong Gan , Weijun Liu , Binzhou Xia","doi":"10.1016/j.jctb.2024.12.003","DOIUrl":"10.1016/j.jctb.2024.12.003","url":null,"abstract":"<div><div>A pair of graphs <math><mo>(</mo><mi>Γ</mi><mo>,</mo><mi>Σ</mi><mo>)</mo></math> is called unstable if their direct product <math><mi>Γ</mi><mo>×</mo><mi>Σ</mi></math> has automorphisms that do not come from <math><mtext>Aut</mtext><mo>(</mo><mi>Γ</mi><mo>)</mo><mo>×</mo><mtext>Aut</mtext><mo>(</mo><mi>Σ</mi><mo>)</mo></math>, and such automorphisms are said to be unexpected. In the special case when <math><mi>Σ</mi><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math>, the stability of <math><mo>(</mo><mi>Γ</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> is well studied in the literature, where the so-called two-fold automorphisms of the graph Γ have played an important role. As a generalization of two-fold automorphisms, the concept of non-diagonal automorphisms was recently introduced to study the stability of general graph pairs. In this paper, we obtain, for a large family of graph pairs, a necessary and sufficient condition to be unstable in terms of the existence of non-diagonal automorphisms. As a byproduct, we determine the stability of graph pairs involving complete graphs or odd cycles, respectively. The former result in fact solves a problem proposed by Dobson, Miklavič and Šparl for undirected graphs, as well as confirms a recent conjecture of Qin, Xia and Zhou.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"171 ","pages":"Pages 140-164"},"PeriodicalIF":1.2,"publicationDate":"2024-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142929318","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Intersecting families with covering number three 与第三个覆盖的家族相交

IF 1.2 1区数学

Journal of Combinatorial Theory Series B Pub Date : 2024-12-18 DOI: 10.1016/j.jctb.2024.12.001

Peter Frankl , Jian Wang

引用次数: 0