Journal of Combinatorial Theory Series B最新文献

{"title":"Invariants of Tutte partitions and a q-analogue","authors":"Eimear Byrne,&nbsp;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}

{"title":"Orientably-regular embeddings of complete multigraphs","authors":"Štefan Gyürki,&nbsp;Soňa Pavlíková,&nbsp;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}

{"title":"On a conjecture of Tokushige for cross-t-intersecting families","authors":"Huajun Zhang ,&nbsp;Biao Wu","doi":"10.1016/j.jctb.2024.11.005","DOIUrl":"10.1016/j.jctb.2024.11.005","url":null,"abstract":"<div><div>Two families of sets <span><math><mi>A</mi></math></span> and <span><math><mi>B</mi></math></span> are called cross-<em>t</em>-intersecting if <span><math><mo>|</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo>|</mo><mo>≥</mo><mi>t</mi></math></span> for all <span><math><mi>A</mi><mo>∈</mo><mi>A</mi></math></span>, <span><math><mi>B</mi><mo>∈</mo><mi>B</mi></math></span>. An active problem in extremal set theory is to determine the maximum product of sizes of cross-<em>t</em>-intersecting families. This incorporates the classical Erdős–Ko–Rado (EKR) problem. In the present paper, we prove that if <span><math><mi>A</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> and <span><math><mi>B</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> are cross-<em>t</em>-intersecting with <span><math><mi>k</mi><mo>≥</mo><mi>t</mi><mo>≥</mo><mn>3</mn></math></span> and <span><math><mi>n</mi><mo>≥</mo><mo>(</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>k</mi><mo>−</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>, then <span><math><mo>|</mo><mi>A</mi><mo>|</mo><mo>|</mo><mi>B</mi><mo>|</mo><mo>≤</mo><msup><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mi>t</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>−</mo><mi>t</mi></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span>. Moreover, equality holds if and only if <span><math><mi>A</mi><mo>=</mo><mi>B</mi></math></span> is a maximum <em>t</em>-intersecting subfamily of <span><math><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></math></span>. This confirms a conjecture of Tokushige for <span><math><mi>t</mi><mo>≥</mo><mn>3</mn></math></span>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"171 ","pages":"Pages 49-70"},"PeriodicalIF":1.2,"publicationDate":"2024-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142929320","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":"Linear three-uniform hypergraphs with no Berge path of given length","authors":"Ervin Győri ,&nbsp;Nika Salia","doi":"10.1016/j.jctb.2024.11.003","DOIUrl":"10.1016/j.jctb.2024.11.003","url":null,"abstract":"<div><div>Extensions of Erdős-Gallai Theorem for general hypergraphs are well studied. In this work, we prove the extension of Erdős-Gallai Theorem for linear hypergraphs. In particular, we show that the number of hyperedges in an <em>n</em>-vertex 3-uniform linear hypergraph, without a Berge path of length <em>k</em> as a subgraph is at most <span><math><mfrac><mrow><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>6</mn></mrow></mfrac><mi>n</mi></math></span> for <span><math><mi>k</mi><mo>≥</mo><mn>4</mn></math></span>. The bound is sharp for infinitely many <em>k</em> and <em>n</em>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"171 ","pages":"Pages 36-48"},"PeriodicalIF":1.2,"publicationDate":"2024-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142929321","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":"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}