{"title":"Counting cycles in planar triangulations","authors":"On-Hei Solomon Lo , Carol T. Zamfirescu","doi":"10.1016/j.jctb.2024.10.002","DOIUrl":"10.1016/j.jctb.2024.10.002","url":null,"abstract":"<div><div>We investigate the minimum number of cycles of specified lengths in planar <em>n</em>-vertex triangulations <em>G</em>. We prove that this number is <span><math><mi>Ω</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> for any cycle length at most <span><math><mn>3</mn><mo>+</mo><mi>max</mi><mo></mo><mo>{</mo><mrow><mi>rad</mi></mrow><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>)</mo><mo>,</mo><mo>⌈</mo><msup><mrow><mo>(</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow><mrow><msub><mrow><mi>log</mi></mrow><mrow><mn>3</mn></mrow></msub><mo></mo><mn>2</mn></mrow></msup><mo>⌉</mo><mo>}</mo></math></span>, where <span><math><mrow><mi>rad</mi></mrow><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>)</mo></math></span> denotes the radius of the triangulation's dual, which is at least logarithmic but can be linear in the order of the triangulation. We also show that there exist planar hamiltonian <em>n</em>-vertex triangulations containing <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> many <em>k</em>-cycles for any <span><math><mi>k</mi><mo>∈</mo><mo>{</mo><mo>⌈</mo><mi>n</mi><mo>−</mo><mroot><mrow><mi>n</mi></mrow><mrow><mn>5</mn></mrow></mroot><mo>⌉</mo><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span>. Furthermore, we prove that planar 4-connected <em>n</em>-vertex triangulations contain <span><math><mi>Ω</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> many <em>k</em>-cycles for every <span><math><mi>k</mi><mo>∈</mo><mo>{</mo><mn>3</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span>, and that, under certain additional conditions, they contain <span><math><mi>Ω</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> <em>k</em>-cycles for many values of <em>k</em>, including <em>n</em>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142586484","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}
{"title":"Trees with many leaves in tournaments","authors":"Alistair Benford , Richard Montgomery","doi":"10.1016/j.jctb.2024.10.001","DOIUrl":"10.1016/j.jctb.2024.10.001","url":null,"abstract":"<div><div>Sumner's universal tournament conjecture states that every <span><math><mo>(</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn><mo>)</mo></math></span>-vertex tournament should contain a copy of every <em>n</em>-vertex oriented tree. If we know the number of leaves of an oriented tree, or its maximum degree, can we guarantee a copy of the tree with fewer vertices in the tournament? Due to work initiated by Häggkvist and Thomason (for number of leaves) and Kühn, Mycroft and Osthus (for maximum degree), it is known that improvements can be made over Sumner's conjecture in some cases, and indeed sometimes an <span><math><mo>(</mo><mi>n</mi><mo>+</mo><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo></math></span>-vertex tournament may be sufficient.</div><div>In this paper, we give new results on these problems. Specifically, we show<ul><li><span>i)</span><span><div>for every <span><math><mi>α</mi><mo>></mo><mn>0</mn></math></span>, there exists <span><math><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><mi>N</mi></math></span> such that, whenever <span><math><mi>n</mi><mo>⩾</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>, every <span><math><mo>(</mo><mo>(</mo><mn>1</mn><mo>+</mo><mi>α</mi><mo>)</mo><mi>n</mi><mo>+</mo><mi>k</mi><mo>)</mo></math></span>-vertex tournament contains a copy of every <em>n</em>-vertex oriented tree with <em>k</em> leaves, and</div></span></li><li><span>ii)</span><span><div>for every <span><math><mi>α</mi><mo>></mo><mn>0</mn></math></span>, there exists <span><math><mi>c</mi><mo>></mo><mn>0</mn></math></span> and <span><math><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><mi>N</mi></math></span> such that, whenever <span><math><mi>n</mi><mo>⩾</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>, every <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>α</mi><mo>)</mo><mi>n</mi></math></span>-vertex tournament contains a copy of every <em>n</em>-vertex oriented tree with maximum degree <span><math><mi>Δ</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>⩽</mo><mi>c</mi><mi>n</mi></math></span>.</div></span></li></ul> Our first result gives an asymptotic form of a conjecture by Havet and Thomassé, while the second improves a result of Mycroft and Naia which applies to trees with polylogarithmic maximum degree.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142442896","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}
Andrzej Dudek , Jarosław Grytczuk , Andrzej Ruciński
{"title":"Erdős-Szekeres type theorems for ordered uniform matchings","authors":"Andrzej Dudek , Jarosław Grytczuk , Andrzej Ruciński","doi":"10.1016/j.jctb.2024.09.004","DOIUrl":"10.1016/j.jctb.2024.09.004","url":null,"abstract":"<div><div>For <span><math><mi>r</mi><mo>,</mo><mi>n</mi><mo>⩾</mo><mn>2</mn></math></span>, an ordered <em>r</em>-uniform matching of size <em>n</em> is an <em>r</em>-uniform hypergraph on a linearly ordered vertex set <em>V</em>, with <span><math><mo>|</mo><mi>V</mi><mo>|</mo><mo>=</mo><mi>r</mi><mi>n</mi></math></span>, consisting of <em>n</em> pairwise disjoint edges. There are <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mn>2</mn><mi>r</mi></mrow></mtd></mtr><mtr><mtd><mi>r</mi></mtd></mtr></mtable><mo>)</mo></mrow></math></span> different ways two edges may intertwine, called here patterns. Among them we identify <span><math><msup><mrow><mn>3</mn></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> collectable patterns <em>P</em>, which have the potential of appearing in arbitrarily large quantities called <em>P</em>-cliques.</div><div>We prove an Erdős-Szekeres type result guaranteeing in <em>every</em> ordered <em>r</em>-uniform matching the presence of a <em>P</em>-clique of a prescribed size, for <em>some</em> collectable pattern <em>P</em>. In particular, in the diagonal case, one of the <em>P</em>-cliques must be of size <span><math><mi>Ω</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><msup><mrow><mn>3</mn></mrow><mrow><mn>1</mn><mo>−</mo><mi>r</mi></mrow></msup></mrow></msup><mo>)</mo></mrow></math></span>. In addition, for <em>each</em> collectable pattern <em>P</em> we show that the largest size of a <em>P</em>-clique in a <em>random</em> ordered <em>r</em>-uniform matching of size <em>n</em> is, with high probability, <span><math><mi>Θ</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>1</mn><mo>/</mo><mi>r</mi></mrow></msup><mo>)</mo></mrow></math></span>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142432825","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}
David Bradley-Williams , Peter J. Cameron , Jan Hubička , Matěj Konečný
{"title":"EPPA numbers of graphs","authors":"David Bradley-Williams , Peter J. Cameron , Jan Hubička , Matěj Konečný","doi":"10.1016/j.jctb.2024.09.003","DOIUrl":"10.1016/j.jctb.2024.09.003","url":null,"abstract":"<div><div>If <em>G</em> is a graph, <em>A</em> and <em>B</em> its induced subgraphs, and <span><math><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></math></span> an isomorphism, we say that <em>f</em> is a <em>partial automorphism</em> of <em>G</em>. In 1992, Hrushovski proved that graphs have the <em>extension property for partial automorphisms</em> (<em>EPPA</em>, also called the <em>Hrushovski property</em>), that is, for every finite graph <em>G</em> there is a finite graph <em>H</em>, an <em>EPPA-witness</em> for <em>G</em>, such that <em>G</em> is an induced subgraph of <em>H</em> and every partial automorphism of <em>G</em> extends to an automorphism of <em>H</em>.</div><div>The <em>EPPA number</em> of a graph <em>G</em>, denoted by <span><math><mrow><mi>eppa</mi></mrow><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, is the smallest number of vertices of an EPPA-witness for <em>G</em>, and we put <span><math><mrow><mi>eppa</mi></mrow><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mi>max</mi><mo></mo><mo>{</mo><mrow><mi>eppa</mi></mrow><mo>(</mo><mi>G</mi><mo>)</mo><mo>:</mo><mo>|</mo><mi>G</mi><mo>|</mo><mo>=</mo><mi>n</mi><mo>}</mo></math></span>. In this note we review the state of the area, prove several lower bounds (in particular, we show that <span><math><mrow><mi>eppa</mi></mrow><mo>(</mo><mi>n</mi><mo>)</mo><mo>≥</mo><mfrac><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow></mfrac></math></span>, thereby identifying the correct base of the exponential) and pose many open questions. We also briefly discuss EPPA numbers of hypergraphs, directed graphs, and <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>-free graphs.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421461","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}
{"title":"Volume rigidity and algebraic shifting","authors":"Denys Bulavka , Eran Nevo , Yuval Peled","doi":"10.1016/j.jctb.2024.09.002","DOIUrl":"10.1016/j.jctb.2024.09.002","url":null,"abstract":"<div><div>We study the generic volume-rigidity of <span><math><mo>(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-dimensional simplicial complexes in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>, and show that the volume-rigidity of a complex can be identified in terms of its exterior shifting. In addition, we establish the volume-rigidity of triangulations of several 2-dimensional surfaces and prove that, in all dimensions >1, volume-rigidity is <em>not</em> characterized by a corresponding hypergraph sparsity property.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142328092","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}
{"title":"Sufficient conditions for perfect mixed tilings","authors":"Eoin Hurley , Felix Joos , Richard Lang","doi":"10.1016/j.jctb.2024.08.007","DOIUrl":"10.1016/j.jctb.2024.08.007","url":null,"abstract":"<div><div>We develop a method to study sufficient conditions for perfect mixed tilings. Our framework allows the embedding of bounded degree graphs <em>H</em> with components of sublinear order. As a corollary, we recover and extend the work of Kühn and Osthus regarding sufficient minimum degree conditions for perfect <em>F</em>-tilings (for an arbitrary fixed graph <em>F</em>) by replacing the <em>F</em>-tiling with the aforementioned graphs <em>H</em>. Moreover, we obtain analogous results for degree sequences and in the setting of uniformly dense graphs. Finally, we asymptotically resolve a conjecture of Komlós in a strong sense.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142314331","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}
Seonghyuk Im , Jaehoon Kim , Younjin Kim , Hong Liu
{"title":"Crux, space constraints and subdivisions","authors":"Seonghyuk Im , Jaehoon Kim , Younjin Kim , Hong Liu","doi":"10.1016/j.jctb.2024.08.005","DOIUrl":"10.1016/j.jctb.2024.08.005","url":null,"abstract":"<div><p>For a given graph <em>H</em>, its subdivisions carry the same topological structure. The existence of <em>H</em>-subdivisions within a graph <em>G</em> has deep connections with topological, structural and extremal properties of <em>G</em>. One prominent example of such a connection, due to Bollobás and Thomason and independently Komlós and Szemerédi, asserts that the average degree of <em>G</em> being <em>d</em> ensures a <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>Ω</mi><mo>(</mo><msqrt><mrow><mi>d</mi></mrow></msqrt><mo>)</mo></mrow></msub></math></span>-subdivision in <em>G</em>. Although this square-root bound is best possible, various results showed that much larger clique subdivisions can be found in a graph for many natural classes. We investigate the connection between crux, a notion capturing the essential order of a graph, and the existence of large clique subdivisions. This reveals the unifying cause underpinning all those improvements for various classes of graphs studied. Roughly speaking, when embedding subdivisions, natural space constraints arise; and such space constraints can be measured via crux.</p><p>Our main result gives an asymptotically optimal bound on the size of a largest clique subdivision in a generic graph <em>G</em>, which is determined by both its average degree and its crux size. As corollaries, we obtain</p><ul><li><span>•</span><span><p>a characterization of extremal graphs for which the square-root bound above is tight: they are essentially disjoint unions of graphs having crux size linear in <em>d</em>;</p></span></li><li><span>•</span><span><p>a unifying approach to find a clique subdivision of almost optimal size in graphs which do not contain a fixed bipartite graph as a subgraph;</p></span></li><li><span>•</span><span><p>and that the clique subdivision size in random graphs <span><math><mi>G</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span> witnesses a dichotomy: when <span><math><mi>p</mi><mo>=</mo><mi>ω</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>)</mo></math></span>, the barrier is the space, while when <span><math><mi>p</mi><mo>=</mo><mi>o</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>)</mo></math></span>, the bottleneck is the density.</p></span></li></ul></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142274290","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}
{"title":"On graph classes with minor-universal elements","authors":"Agelos Georgakopoulos","doi":"10.1016/j.jctb.2024.09.001","DOIUrl":"10.1016/j.jctb.2024.09.001","url":null,"abstract":"<div><p>A graph <em>U</em> is universal for a graph class <span><math><mi>C</mi><mo>∋</mo><mi>U</mi></math></span>, if every <span><math><mi>G</mi><mo>∈</mo><mi>C</mi></math></span> is a minor of <em>U</em>. We prove the existence or absence of universal graphs in several natural graph classes, including graphs component-wise embeddable into a surface, and graphs forbidding <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span>, or <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>3</mn></mrow></msub></math></span>, or <span><math><msub><mrow><mi>K</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span> as a minor. We prove the existence of uncountably many minor-closed classes of countable graphs that do not have a universal element.</p><p>Some of our results and questions may be of interest from the finite graph perspective. In particular, one of our side-results is that every <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span>-minor-free graph is a minor of a <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span>-minor-free graph of maximum degree 22.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0095895624000741/pdfft?md5=b5bdf1f35e156e3581f5a8ffea761652&pid=1-s2.0-S0095895624000741-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142240971","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}
{"title":"Lift theorems for representations of matroids over pastures","authors":"Matthew Baker , Oliver Lorscheid","doi":"10.1016/j.jctb.2024.08.004","DOIUrl":"10.1016/j.jctb.2024.08.004","url":null,"abstract":"<div><p>Pastures are a class of field-like algebraic objects which include both partial fields and hyperfields and have nice categorical properties. We prove several lift theorems for representations of matroids over pastures, including a generalization of Pendavingh and van Zwam's Lift Theorem for partial fields. By embedding the earlier theory into a more general framework, we are able to establish new results even in the case of lifts of partial fields, for example the conjecture of Pendavingh–van Zwam that their lift construction is idempotent. We give numerous applications to matroid representations, e.g. we show that, up to projective equivalence, every pair consisting of a hexagonal representation and an orientation lifts uniquely to a near-regular representation. The proofs are different from the arguments used by Pendavingh and van Zwam, relying instead on a result of Gelfand–Rybnikov–Stone inspired by Tutte's homotopy theorem.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142158090","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}
Louis Esperet , Ugo Giocanti , Clément Legrand-Duchesne
{"title":"The structure of quasi-transitive graphs avoiding a minor with applications to the domino problem","authors":"Louis Esperet , Ugo Giocanti , Clément Legrand-Duchesne","doi":"10.1016/j.jctb.2024.08.002","DOIUrl":"10.1016/j.jctb.2024.08.002","url":null,"abstract":"<div><p>An infinite graph is quasi-transitive if its vertex set has finitely many orbits under the action of its automorphism group. In this paper we obtain a structure theorem for locally finite quasi-transitive graphs avoiding a minor, which is reminiscent of the Robertson-Seymour Graph Minor Structure Theorem. We prove that every locally finite quasi-transitive graph <em>G</em> avoiding a minor has a tree-decomposition whose torsos are finite or planar; moreover the tree-decomposition is canonical, i.e. invariant under the action of the automorphism group of <em>G</em>. As applications of this result, we prove the following.</p><ul><li><span>•</span><span><p>Every locally finite quasi-transitive graph attains its Hadwiger number, that is, if such a graph contains arbitrarily large clique minors, then it contains an infinite clique minor. This extends a result of Thomassen (1992) <span><span>[38]</span></span> who proved it in the (quasi-)4-connected case and suggested that this assumption could be omitted. In particular, this shows that a Cayley graph excludes a finite minor if and only if it avoids the countable clique as a minor.</p></span></li><li><span>•</span><span><p>Locally finite quasi-transitive graphs avoiding a minor are accessible (in the sense of Thomassen and Woess), which extends known results on planar graphs to any proper minor-closed family.</p></span></li><li><span>•</span><span><p>Minor-excluded finitely generated groups are accessible (in the group-theoretic sense) and finitely presented, which extends classical results on planar groups.</p></span></li><li><span>•</span><span><p>The domino problem is decidable in a minor-excluded finitely generated group if and only if the group is virtually free, which proves the minor-excluded case of a conjecture of Ballier and Stein (2018) <span><span>[7]</span></span>.</p></span></li></ul></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142121970","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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