Pub Date : 2025-01-13DOI: 10.1016/j.jctb.2024.12.007
Nikolai Terekhov , Maksim Zhukovskii
The weak saturation number 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 such that . Our lower bounds imply that all values in the interval with step size are achievable by for graphs F with minimum degree δ (while any value outside this interval is not achievable).
{"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 <span><math><mrow><mi>wsat</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> is the minimum number of edges in a graph on <em>n</em> vertices such that all the missing edges can be activated sequentially so that each new edge creates a copy of <em>F</em>. 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 <em>F</em>, there exists <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>F</mi></mrow></msub></math></span> such that <span><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></span>. Our lower bounds imply that all values in the interval <span><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></span> with step size <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>δ</mi><mo>+</mo><mn>1</mn></mrow></mfrac></math></span> are achievable by <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>F</mi></mrow></msub></math></span> for graphs <em>F</em> with minimum degree <em>δ</em> (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}
Pub Date : 2025-01-13DOI: 10.1016/j.jctb.2024.12.009
Zach Hunter , Aleksa Milojević , Benny Sudakov , István Tomon
The celebrated Kővári-Sós-Turán theorem states that any n-vertex graph containing no copy of the complete bipartite graph has at most 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 -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 -free subgraphs with large degree and bounding the average degree of -free graphs which do not contain induced subdivisions of a fixed graph.
{"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 <em>n</em>-vertex graph containing no copy of the complete bipartite graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>s</mi></mrow></msub></math></span> has at most <span><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></span> 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 <em>H</em> is a bipartite graph, then an induced <em>H</em>-free and <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>s</mi></mrow></msub></math></span>-free graph cannot have much more edges than an <em>H</em>-free graph. We provide evidence for this conjecture by considering trees, cycles, the cube graph, and bipartite graphs with degrees bounded by <em>k</em> 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 <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-free subgraphs with large degree and bounding the average degree of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>s</mi></mrow></msub></math></span>-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}
We prove that there exists a function such that every directed graph G contains either k directed odd cycles where every vertex of G is contained in at most two of them, or a set of at most vertices meeting all directed odd cycles. We give a polynomial-time algorithm for fixed k which outputs one of the two outcomes. This extends the half-integral Erdős-Pósa theorem for undirected odd cycles by Reed [Combinatorica 1999] to directed graphs.
{"title":"A half-integral Erdős-Pósa theorem for directed odd cycles","authors":"Ken-ichi Kawarabayashi , Stephan Kreutzer , O-joung Kwon , Qiqin Xie","doi":"10.1016/j.jctb.2024.12.008","DOIUrl":"10.1016/j.jctb.2024.12.008","url":null,"abstract":"<div><div>We prove that there exists a function <span><math><mi>f</mi><mo>:</mo><mi>N</mi><mo>→</mo><mi>R</mi></math></span> such that every directed graph <em>G</em> contains either <em>k</em> directed odd cycles where every vertex of <em>G</em> is contained in at most two of them, or a set of at most <span><math><mi>f</mi><mo>(</mo><mi>k</mi><mo>)</mo></math></span> vertices meeting all directed odd cycles. We give a polynomial-time algorithm for fixed <em>k</em> which outputs one of the two outcomes. This extends the half-integral Erdős-Pósa theorem for undirected odd cycles by Reed [Combinatorica 1999] to directed graphs.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"172 ","pages":"Pages 115-145"},"PeriodicalIF":1.2,"publicationDate":"2025-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143092867","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}
Pub Date : 2024-12-31DOI: 10.1016/j.jctb.2024.12.005
László Pyber , Saveliy V. Skresanov
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 for some universal constant , unless the graph is a Johnson or Hamming graph. We prove that the motion of a distance-regular graph of diameter on n vertices is at least for some universal constant , 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 , where 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.
{"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 <em>n</em> vertices of diameter greater than two is at least <span><math><mi>n</mi><mo>/</mo><mi>C</mi></math></span> for some universal constant <span><math><mi>C</mi><mo>></mo><mn>0</mn></math></span>, unless the graph is a Johnson or Hamming graph. We prove that the motion of a distance-regular graph of diameter <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span> on <em>n</em> vertices is at least <span><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></span> for some universal constant <span><math><mi>C</mi><mo>></mo><mn>0</mn></math></span>, 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 <span><math><mi>n</mi><mo>/</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span>, where <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> depends exponentially on <em>d</em>. 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}
Pub Date : 2024-12-30DOI: 10.1016/j.jctb.2024.12.004
Patrick Hompe, Tony Huynh
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 .
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 , there exists a constant 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 .
{"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 <em>G</em> is a simple <em>n</em>-vertex edge-colored graph with <em>n</em> color classes of size at least <em>r</em>, then <em>G</em> contains a rainbow cycle of length at most <span><math><mo>⌈</mo><mi>n</mi><mo>/</mo><mi>r</mi><mo>⌉</mo></math></span>.</div><div>In this paper, we prove that, for fixed <em>r</em>, Aharoni's conjecture holds up to an additive constant. Specifically, we show that for each fixed <span><math><mi>r</mi><mo>⩾</mo><mn>1</mn></math></span>, there exists a constant <span><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></span> such that if <em>G</em> is a simple <em>n</em>-vertex edge-colored graph with <em>n</em> color classes of size at least <em>r</em>, then <em>G</em> contains a rainbow cycle of length at most <span><math><mi>n</mi><mo>/</mo><mi>r</mi><mo>+</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span>.</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}
Pub Date : 2024-12-27DOI: 10.1016/j.jctb.2024.12.006
David Fabian , Patrick Morris , Tibor Szabó
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 , and its dependence on properties of the graph H. Our focus is on H which define relatively fast bootstrap processes, that is, with being at most linear in n. We study the graph class of trees, showing that one can bound by a quadratic function in 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.
{"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 <em>H</em> and an <em>n</em>-vertex graph <em>G</em>, the <em>H</em>-bootstrap process on <em>G</em> is the process which starts with <em>G</em> and, at every time step, adds any missing edges on the vertices of <em>G</em> that complete a copy of <em>H</em>. This process eventually stabilises and we are interested in the extremal question raised by Bollobás of determining the maximum <em>running time</em> (number of time steps before stabilising) of this process over all possible choices of <em>n</em>-vertex graph <em>G</em>. In this paper, we initiate a systematic study of the asymptotics of this parameter, denoted <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>, and its dependence on properties of the graph <em>H</em>. Our focus is on <em>H</em> which define relatively fast bootstrap processes, that is, with <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> being at most linear in <em>n</em>. We study the graph class of trees, showing that one can bound <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> by a quadratic function in <span><math><mi>v</mi><mo>(</mo><mi>T</mi><mo>)</mo></math></span> for all trees <em>T</em> and all <em>n</em>. We then go on to explore the relationship between the running time of the <em>H</em>-process and the minimum vertex degree and connectivity of <em>H</em>.</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}
Pub Date : 2024-12-19DOI: 10.1016/j.jctb.2024.12.003
Yunsong Gan , Weijun Liu , Binzhou Xia
A pair of graphs is called unstable if their direct product has automorphisms that do not come from , and such automorphisms are said to be unexpected. In the special case when , the stability of 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.
{"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 <span><math><mo>(</mo><mi>Γ</mi><mo>,</mo><mi>Σ</mi><mo>)</mo></math></span> is called unstable if their direct product <span><math><mi>Γ</mi><mo>×</mo><mi>Σ</mi></math></span> has automorphisms that do not come from <span><math><mtext>Aut</mtext><mo>(</mo><mi>Γ</mi><mo>)</mo><mo>×</mo><mtext>Aut</mtext><mo>(</mo><mi>Σ</mi><mo>)</mo></math></span>, and such automorphisms are said to be unexpected. In the special case when <span><math><mi>Σ</mi><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, the stability of <span><math><mo>(</mo><mi>Γ</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> 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}
Pub Date : 2024-12-18DOI: 10.1016/j.jctb.2024.12.001
Peter Frankl , Jian Wang
We consider k-graphs on n vertices, that is, . A k-graph is called intersecting if for all . In the present paper we prove that for , , any intersecting k-graph with covering number at least three, satisfies , the best possible upper bound which was proved in [4] subject to exponential constraints .
{"title":"Intersecting families with covering number three","authors":"Peter Frankl , Jian Wang","doi":"10.1016/j.jctb.2024.12.001","DOIUrl":"10.1016/j.jctb.2024.12.001","url":null,"abstract":"<div><div>We consider <em>k</em>-graphs on <em>n</em> vertices, that is, <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>. A <em>k</em>-graph <span><math><mi>F</mi></math></span> is called intersecting if <span><math><mi>F</mi><mo>∩</mo><msup><mrow><mi>F</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>≠</mo><mo>∅</mo></math></span> for all <span><math><mi>F</mi><mo>,</mo><msup><mrow><mi>F</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi>F</mi></math></span>. In the present paper we prove that for <span><math><mi>k</mi><mo>≥</mo><mn>7</mn></math></span>, <span><math><mi>n</mi><mo>≥</mo><mn>2</mn><mi>k</mi></math></span>, any intersecting <em>k</em>-graph <span><math><mi>F</mi></math></span> with covering number at least three, satisfies <span><math><mo>|</mo><mi>F</mi><mo>|</mo><mo>≤</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mi>k</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mn>2</mn><mi>k</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mi>k</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>−</mo><mn>3</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mn>3</mn></math></span>, the best possible upper bound which was proved in <span><span>[4]</span></span> subject to exponential constraints <span><math><mi>n</mi><mo>></mo><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"171 ","pages":"Pages 96-139"},"PeriodicalIF":1.2,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142929317","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}
Pub Date : 2024-12-18DOI: 10.1016/j.jctb.2024.12.002
Eimear Byrne, Andrew Fulcher
We describe a construction of the Tutte polynomial for both matroids and q-matroids based on an appropriate partition of the underlying support lattice into intervals that correspond to prime-free minors, which we call a Tutte partition. We show that such partitions in the matroid case include the class of partitions arising in Crapo's definition of the Tutte polynomial, while not representing a direct q-analogue of such partitions. We propose axioms of a q-Tutte-Grothendieck invariant and show that this yields a q-analogue of a Tutte-Grothendieck invariant. We establish the connection between the rank generating polynomial and the Tutte polynomial, showing that one can be obtained from the other by convolution.
{"title":"Invariants of Tutte partitions and a q-analogue","authors":"Eimear Byrne, Andrew Fulcher","doi":"10.1016/j.jctb.2024.12.002","DOIUrl":"10.1016/j.jctb.2024.12.002","url":null,"abstract":"<div><div>We describe a construction of the Tutte polynomial for both matroids and <em>q</em>-matroids based on an appropriate partition of the underlying support lattice into intervals that correspond to prime-free minors, which we call a Tutte partition. We show that such partitions in the matroid case include the class of partitions arising in Crapo's definition of the Tutte polynomial, while not representing a direct <em>q</em>-analogue of such partitions. We propose axioms of a <em>q</em>-Tutte-Grothendieck invariant and show that this yields a <em>q</em>-analogue of a Tutte-Grothendieck invariant. We establish the connection between the rank generating polynomial and the Tutte polynomial, showing that one can be obtained from the other by convolution.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"172 ","pages":"Pages 1-43"},"PeriodicalIF":1.2,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143092945","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}
Pub Date : 2024-12-11DOI: 10.1016/j.jctb.2024.11.004
Štefan Gyürki, Soňa Pavlíková, Jozef Širáň
An embedding of a graph on an orientable surface is orientably-regular (or rotary, in an equivalent terminology) if the group of orientation-preserving automorphisms of the embedding is transitive (and hence regular) on incident vertex-edge pairs of the graph. A classification of orientably-regular embeddings of complete graphs was obtained by L.D. James and G.A. Jones (1985) [10], pointing out interesting connections to finite fields and Frobenius groups. By a combination of graph-theoretic methods and tools from combinatorial group theory we extend results of James and Jones to classification of orientably-regular embeddings of complete multigraphs with arbitrary edge-multiplicity.
{"title":"Orientably-regular embeddings of complete multigraphs","authors":"Štefan Gyürki, Soňa Pavlíková, Jozef Širáň","doi":"10.1016/j.jctb.2024.11.004","DOIUrl":"10.1016/j.jctb.2024.11.004","url":null,"abstract":"<div><div>An embedding of a graph on an orientable surface is <em>orientably-regular</em> (or <em>rotary</em>, in an equivalent terminology) if the group of orientation-preserving automorphisms of the embedding is transitive (and hence regular) on incident vertex-edge pairs of the graph. A classification of orientably-regular embeddings of complete graphs was obtained by L.D. James and G.A. Jones (1985) <span><span>[10]</span></span>, pointing out interesting connections to finite fields and Frobenius groups. By a combination of graph-theoretic methods and tools from combinatorial group theory we extend results of James and Jones to classification of orientably-regular embeddings of complete multigraphs with arbitrary edge-multiplicity.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"171 ","pages":"Pages 71-95"},"PeriodicalIF":1.2,"publicationDate":"2024-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142929316","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}
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