{"title":"A co-preLie structure from chronological loop erasure in graph walks","authors":"Loïc Foissy , Pierre-Louis Giscard , Cécile Mammez","doi":"10.1016/j.ejc.2024.103967","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103967","url":null,"abstract":"<div><p>We show that the chronological removal of cycles from a walk on a graph, known as Lawler’s loop-erasing procedure, generates a preLie co-algebra on the vector space spanned by the walks. In addition, we prove that the tensor and symmetric algebras of graph walks are Hopf algebras, provide their antipodes explicitly and recover the preLie co-algebra from a brace coalgebra on the tensor algebra of graph walks. Finally we exhibit sub-Hopf algebras associated to particular types of walks.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824000520/pdfft?md5=2816f59bd5b837c134bedca21dd85a3f&pid=1-s2.0-S0195669824000520-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140549963","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On sum-intersecting families of positive integers","authors":"Aaron Berger, Nitya Mani","doi":"10.1016/j.ejc.2024.103963","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103963","url":null,"abstract":"<div><p>We study the following natural arithmetic question regarding intersecting families: how large can a family of subsets of integers from <span><math><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mi>n</mi><mo>}</mo></mrow></math></span> be such that, for every pair of subsets in the family, the intersection contains a <em>sum</em> <span><math><mrow><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mi>z</mi></mrow></math></span>? We conjecture that any such <em>sum-intersecting</em> family must have size at most <span><math><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mi>⋅</mi><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span> (which would be tight if correct). Towards this conjecture, we show that every sum-intersecting family has at most <span><math><mrow><mn>0</mn><mo>.</mo><mn>32</mn><mi>⋅</mi><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span> subsets.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140543870","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A note on non-empty cross-intersecting families","authors":"Menglong Zhang, Tao Feng","doi":"10.1016/j.ejc.2024.103968","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103968","url":null,"abstract":"<div><p>The families <span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⊆</mo><mfenced><mrow><mfrac><mrow><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⊆</mo><mfenced><mrow><mfrac><mrow><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>⊆</mo><mfenced><mrow><mfrac><mrow><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></mfrac></mrow></mfenced></mrow></math></span> are said to be cross-intersecting if <span><math><mrow><mrow><mo>|</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∩</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>|</mo></mrow><mo>⩾</mo><mn>1</mn></mrow></math></span> for any <span><math><mrow><mn>1</mn><mo>⩽</mo><mi>i</mi><mo><</mo><mi>j</mi><mo>⩽</mo><mi>r</mi></mrow></math></span> and <span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></math></span>, <span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></math></span>. Cross-intersecting families <span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></math></span> are said to be <em>non-empty</em> if <span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≠</mo><mo>0̸</mo></mrow></math></span> for any <span><math><mrow><mn>1</mn><mo>⩽</mo><mi>i</mi><mo>⩽</mo><mi>r</mi></mrow></math></span>. This paper shows that if <span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⊆</mo><mfenced><mrow><mfrac><mrow><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⊆</mo><mfenced><mrow><mfrac><mrow><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>⊆</mo><mfenced><mrow><mfrac><mrow><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></mfrac></mrow></mfenced></mrow></math></span> are non-empty cross-intersecting families with <span><math><mrow><ms","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140535953","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Nearly extremal non-trivial cross t-intersecting families and r-wise t-intersecting families","authors":"Mengyu Cao , Mei Lu , Benjian Lv , Kaishun Wang","doi":"10.1016/j.ejc.2024.103958","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103958","url":null,"abstract":"<div><p>Let <span><math><mi>n</mi></math></span>, <span><math><mi>r</mi></math></span>, <span><math><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></math></span> and <span><math><mi>t</mi></math></span> be positive integers with <span><math><mrow><mi>r</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, and <span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub><mspace></mspace><mrow><mo>(</mo><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>r</mi><mo>)</mo></mrow></mrow></math></span> a family of <span><math><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>-subsets of an <span><math><mi>n</mi></math></span>-set <span><math><mi>V</mi></math></span>. The families <span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mspace></mspace><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></math></span> are said to be <span><math><mi>r</mi></math></span>-cross <span><math><mi>t</mi></math></span>-intersecting if <span><math><mrow><mrow><mo>|</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∩</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∩</mo><mo>⋯</mo><mo>∩</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>|</mo></mrow><mo>≥</mo><mi>t</mi></mrow></math></span> for all <span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub><mspace></mspace><mrow><mo>(</mo><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>r</mi><mo>)</mo></mrow><mo>,</mo></mrow></math></span> and said to be non-trivial if <span><math><mrow><mrow><mo>|</mo><msub><mrow><mo>∩</mo></mrow><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>r</mi></mrow></msub><msub><mrow><mo>∩</mo></mrow><mrow><mi>F</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub><mi>F</mi><mo>|</mo></mrow><mo><</mo><mi>t</mi></mrow></math></span>. If the <span><math><mi>r</mi></math></span>-cross <span><math><mi>t</mi></math></span>-intersecting families <span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></math></span> satisfy <span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mo>⋯</mo><mo>=</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>=</mo><mi>F</mi></mrow></math></span>, then <span><math><mi>F</mi></math></span> is well known as <span><math><mi>r</mi></math></span>-wise <span><math><mi>t</mi></math></span>-intersecting. In this paper, we first describe the structure of maximal 2-cross <span><math><mi>t</mi></math></span>-intersecting families with given <span><math><mi>t</mi></math></s","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140343891","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The binomial-Stirling–Eulerian polynomials","authors":"Kathy Q. Ji , Zhicong Lin","doi":"10.1016/j.ejc.2024.103962","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103962","url":null,"abstract":"<div><p>We introduce the binomial-Stirling–Eulerian polynomials, denoted <span><math><mrow><msub><mrow><mover><mrow><mi>A</mi></mrow><mrow><mo>̃</mo></mrow></mover></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>|</mo><mi>α</mi><mo>)</mo></mrow></mrow></math></span>, which encompass binomial coefficients, Eulerian numbers and two Stirling statistics: the left-to-right minima and the right-to-left minima. When <span><math><mrow><mi>α</mi><mo>=</mo><mn>1</mn></mrow></math></span>, these polynomials reduce to the binomial-Eulerian polynomials <span><math><mrow><msub><mrow><mover><mrow><mi>A</mi></mrow><mrow><mo>̃</mo></mrow></mover></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></math></span>, originally named by Shareshian and Wachs and explored by Chung–Graham–Knuth and Postnikov–Reiner–Williams. We investigate the <span><math><mi>γ</mi></math></span>-positivity of <span><math><mrow><msub><mrow><mover><mrow><mi>A</mi></mrow><mrow><mo>̃</mo></mrow></mover></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>|</mo><mi>α</mi><mo>)</mo></mrow></mrow></math></span> from two aspects: <span><math><mo>•</mo></math></span> firstly by employing the grammatical calculus introduced by Chen; <span><math><mo>•</mo></math></span> and secondly by constructing a new group action on permutations. These results extend the symmetric Eulerian identity found by Chung, Graham and Knuth, and the <span><math><mi>γ</mi></math></span>-positivity of <span><math><mrow><msub><mrow><mover><mrow><mi>A</mi></mrow><mrow><mo>̃</mo></mrow></mover></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></math></span> first demonstrated by Postnikov, Reiner and Williams.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140343889","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Maria Axenovich , António Girão , Lawrence Hollom , Julien Portier , Emil Powierski , Michael Savery , Youri Tamitegama , Leo Versteegen
{"title":"A note on interval colourings of graphs","authors":"Maria Axenovich , António Girão , Lawrence Hollom , Julien Portier , Emil Powierski , Michael Savery , Youri Tamitegama , Leo Versteegen","doi":"10.1016/j.ejc.2024.103956","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103956","url":null,"abstract":"<div><p>A graph is said to be <em>interval colourable</em> if it admits a proper edge-colouring using palette <span><math><mi>N</mi></math></span> in which the set of colours of edges that are incident to each vertex is an interval. The <em>interval colouring thickness</em> of a graph <span><math><mi>G</mi></math></span> is the minimum <span><math><mi>k</mi></math></span> such that <span><math><mi>G</mi></math></span> can be edge-decomposed into <span><math><mi>k</mi></math></span> interval colourable graphs. We show that <span><math><mrow><mi>θ</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span>, the maximum interval colouring thickness of an <span><math><mi>n</mi></math></span>-vertex graph, satisfies <span><math><mrow><mi>θ</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><mi>Ω</mi><mrow><mo>(</mo><mo>log</mo><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>/</mo><mo>log</mo><mo>log</mo><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>θ</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>⩽</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>5</mn><mo>/</mo><mn>6</mn><mo>+</mo><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msup></mrow></math></span>, which improves on the trivial lower bound and the upper bound given by the first author and Zheng. As a corollary, we answer a question of Asratian, Casselgren, and Petrosyan and disprove a conjecture of Borowiecka-Olszewska, Drgas-Burchardt, Javier-Nol, and Zuazua. We also confirm a conjecture of the first author that any interval colouring of an <span><math><mi>n</mi></math></span>-vertex planar graph uses at most <span><math><mrow><mn>3</mn><mi>n</mi><mo>/</mo><mn>2</mn><mo>−</mo><mn>2</mn></mrow></math></span> colours.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824000416/pdfft?md5=f4b4a879ad13e34948a7eab92d5e024c&pid=1-s2.0-S0195669824000416-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140343890","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Hecke-type series involving infinite products","authors":"Bing He","doi":"10.1016/j.ejc.2024.103959","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103959","url":null,"abstract":"<div><p>In this paper, we study Hecke-type series involving infinite products. In particular, we establish some Hecke-type series involving infinite products and then obtain truncated versions of these series as well as truncated forms of some other known series of such types. Finally, as an application, we deduce six infinite families of inequalities for various partition functions. Our proofs of the main results heavily rely on a formula from the work of Liu (2013).</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140339761","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Bounding clique size in squares of planar graphs","authors":"Daniel W. Cranston","doi":"10.1016/j.ejc.2024.103960","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103960","url":null,"abstract":"<div><p>Wegner conjectured that if <span><math><mi>G</mi></math></span> is a planar graph with maximum degree <span><math><mrow><mi>Δ</mi><mo>≥</mo><mn>8</mn></mrow></math></span>, then <span><math><mrow><mi>χ</mi><mrow><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mo>≤</mo><mfenced><mrow><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi>Δ</mi></mrow></mfenced><mo>+</mo><mn>1</mn></mrow></math></span>. This problem has received much attention, but remains open for all <span><math><mrow><mi>Δ</mi><mo>≥</mo><mn>8</mn></mrow></math></span>. Here we prove an analogous bound on <span><math><mrow><mi>ω</mi><mrow><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span>: If <span><math><mi>G</mi></math></span> is a plane graph with <span><math><mrow><mi>Δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mn>36</mn></mrow></math></span>, then <span><math><mrow><mi>ω</mi><mrow><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mo>≤</mo><mrow><mo>⌊</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi>Δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>⌋</mo></mrow><mo>+</mo><mn>1</mn></mrow></math></span>. In fact, this is a corollary of the following lemma, which is our main result. If <span><math><mi>G</mi></math></span> is a plane graph with <span><math><mrow><mi>Δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mn>19</mn></mrow></math></span> and <span><math><mi>S</mi></math></span> is a maximal clique in <span><math><msup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> with <span><math><mrow><mrow><mo>|</mo><mi>S</mi><mo>|</mo></mrow><mo>≥</mo><mi>Δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mn>20</mn></mrow></math></span>, then there exist <span><math><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> such that <span><math><mrow><mi>S</mi><mo>=</mo><mrow><mo>{</mo><mi>w</mi><mo>:</mo><mrow><mo>|</mo><mi>N</mi><mrow><mo>[</mo><mi>w</mi><mo>]</mo></mrow><mo>∩</mo><mrow><mo>{</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>}</mo></mrow><mo>|</mo></mrow><mo>≥</mo><mn>2</mn><mo>}</mo></mrow></mrow></math></span>.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140332859","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Bounding the distant irregularity strength of graphs via a non-uniformly biased random weight assignment","authors":"Jakub Przybyło","doi":"10.1016/j.ejc.2024.103961","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103961","url":null,"abstract":"<div><p>Given an edge <span><math><mi>k</mi></math></span>-weighting <span><math><mrow><mi>ω</mi><mo>:</mo><mi>E</mi><mo>→</mo><mrow><mo>[</mo><mi>k</mi><mo>]</mo></mrow></mrow></math></span> of a graph <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span>, the weighted degree of a vertex <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi></mrow></math></span> is the sum of its incident weights. The least <span><math><mi>k</mi></math></span> for which there exists an edge <span><math><mi>k</mi></math></span>-weighting such that the resulting weighted degrees of the vertices at distance at most <span><math><mi>r</mi></math></span> in <span><math><mi>G</mi></math></span> are distinct is called the <span><math><mi>r</mi></math></span>-distant irregularity strength, and denoted <span><math><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. This concept links the well-known 1–2–3 Conjecture, corresponding to <span><math><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, with the irregularity strength of graphs, <span><math><mrow><mi>s</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, which coincides with <span><math><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> for every <span><math><mi>r</mi></math></span> at least the diameter of <span><math><mi>G</mi></math></span>. It is believed that for every <span><math><mrow><mi>r</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, <span><math><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>)</mo></mrow><msup><mrow><mi>Δ</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span>, where <span><math><mi>Δ</mi></math></span> is the maximum degree of <span><math><mi>G</mi></math></span>, while it is known that <span><math><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>6</mn><msup><mrow><mi>Δ</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span> in general and <span><math><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mrow><mo>(</mo><mn>4</mn><mo>+</mo><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>)</mo></mrow><msup><mrow><mi>Δ</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span> for graphs with minimum degree <span><math><mi>δ</mi></math></span> at least <span><math><mrow><msup><mrow><mo>log</mo></mrow><mrow><mn>8</mn></mrow></msup><mi>Δ</mi></mrow></math></span>. We apply the probabilistic method i","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140296879","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Exact results on generalized Erdős-Gallai problems","authors":"Debsoumya Chakraborti , Da Qi Chen","doi":"10.1016/j.ejc.2024.103955","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103955","url":null,"abstract":"<div><p>Generalized Turán problems have been a central topic of study in extremal combinatorics throughout the last few decades. One such problem is maximizing the number of cliques of size <span><math><mi>t</mi></math></span> in a graph of a fixed order that does not contain any path (or cycle) of length at least a given number. Both of the path-free and cycle-free extremal problems were recently considered and asymptotically solved by Luo. We fully resolve these problems by characterizing all possible extremal graphs. We further extend these results by solving the edge-variant of these problems where the number of edges is fixed instead of the number of vertices. We similarly obtain exact characterization of the extremal graphs for these edge variants.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824000404/pdfft?md5=bd17e91f57524428831c3b9e24030540&pid=1-s2.0-S0195669824000404-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140290763","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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