Journal of Combinatorial Theory Series B最新文献

{"title":"Note on disjoint faces in simple topological graphs","authors":"Ji Zeng","doi":"10.1016/j.jctb.2024.11.002","DOIUrl":"10.1016/j.jctb.2024.11.002","url":null,"abstract":"<div><div>We prove that every <em>n</em>-vertex complete simple topological graph generates at least <span><math><mi>Ω</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> pairwise disjoint 4-faces. This improves upon a recent result by Hubard and Suk. As an immediate corollary, every <em>n</em>-vertex complete simple topological graph drawn in the unit square generates a 4-face with area at most <span><math><mi>O</mi><mo>(</mo><mn>1</mn><mo>/</mo><mi>n</mi><mo>)</mo></math></span>. This can be seen as a topological variant of the Heilbronn problem for quadrilaterals. We construct examples showing that our result is asymptotically tight. We also discuss the similar problem for <em>k</em>-faces with arbitrary <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"171 ","pages":"Pages 28-35"},"PeriodicalIF":1.2,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142743022","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":"A characterization of the Grassmann graphs","authors":"Alexander L. Gavrilyuk ,&nbsp;Jack H. Koolen","doi":"10.1016/j.jctb.2024.11.001","DOIUrl":"10.1016/j.jctb.2024.11.001","url":null,"abstract":"<div><div>The Grassmann graph <span><math><msub><mrow><mi>J</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>D</mi><mo>)</mo></math></span> is a graph on the <em>D</em>-dimensional subspaces of <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> with two subspaces being adjacent if their intersection has dimension <span><math><mi>D</mi><mo>−</mo><mn>1</mn></math></span>. Characterizing these graphs by their intersection numbers is an important step towards a solution of the classification problem for <span><math><mo>(</mo><mi>P</mi><mrow><mspace></mspace><mi>and</mi><mspace></mspace></mrow><mi>Q</mi><mo>)</mo></math></span>-polynomial association schemes, posed by Bannai and Ito in their monograph “Algebraic Combinatorics I” (1984).</div><div>Metsch (1995) <span><span>[37]</span></span> showed that the Grassmann graph <span><math><msub><mrow><mi>J</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>D</mi><mo>)</mo></math></span> with <span><math><mi>D</mi><mo>≥</mo><mn>3</mn></math></span> is characterized by its intersection numbers except for the following two principal open cases: <span><math><mi>n</mi><mo>=</mo><mn>2</mn><mi>D</mi></math></span> or <span><math><mi>n</mi><mo>=</mo><mn>2</mn><mi>D</mi><mo>+</mo><mn>1</mn></math></span>. Van Dam and Koolen (2005) <span><span>[57]</span></span> constructed the twisted Grassmann graphs with the same intersection numbers as the Grassmann graphs <span><math><msub><mrow><mi>J</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mn>2</mn><mi>D</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>D</mi><mo>)</mo></math></span> (for any prime power <em>q</em> and <span><math><mi>D</mi><mo>≥</mo><mn>2</mn></math></span>), but not isomorphic to the latter ones. This shows that characterizing the graphs in the remaining cases would require a conceptually new approach.</div><div>We prove that the Grassmann graph <span><math><msub><mrow><mi>J</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mn>2</mn><mi>D</mi><mo>,</mo><mi>D</mi><mo>)</mo></math></span> is characterized by its intersection numbers provided that <em>D</em> is large enough.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"171 ","pages":"Pages 1-27"},"PeriodicalIF":1.2,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142663746","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":"Counting cycles in planar triangulations","authors":"On-Hei Solomon Lo ,&nbsp;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":"170 ","pages":"Pages 335-351"},"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 ,&nbsp;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>&gt;</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>&gt;</mo><mn>0</mn></math></span>, there exists <span><math><mi>c</mi><mo>&gt;</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":"170 ","pages":"Pages 260-334"},"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}

{"title":"Erdős-Szekeres type theorems for ordered uniform matchings","authors":"Andrzej Dudek ,&nbsp;Jarosław Grytczuk ,&nbsp;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":"170 ","pages":"Pages 225-259"},"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}

{"title":"EPPA numbers of graphs","authors":"David Bradley-Williams ,&nbsp;Peter J. Cameron ,&nbsp;Jan Hubička ,&nbsp;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":"170 ","pages":"Pages 203-224"},"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 ,&nbsp;Eran Nevo ,&nbsp;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 &gt;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":"170 ","pages":"Pages 189-202"},"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 ,&nbsp;Felix Joos ,&nbsp;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":"170 ","pages":"Pages 128-188"},"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}

{"title":"Crux, space constraints and subdivisions","authors":"Seonghyuk Im ,&nbsp;Jaehoon Kim ,&nbsp;Younjin Kim ,&nbsp;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":"170 ","pages":"Pages 82-127"},"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":"170 ","pages":"Pages 56-81"},"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}