European Journal of Combinatorics最新文献

{"title":"Corrigendum to “Describing quasi-graphic matroids” [European J. Combin. 85 (2020) 103062]","authors":"","doi":"10.1016/j.ejc.2024.104004","DOIUrl":"10.1016/j.ejc.2024.104004","url":null,"abstract":"","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"122 ","pages":"Article 104004"},"PeriodicalIF":1.0,"publicationDate":"2024-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824000891/pdfft?md5=8f6dba2471a3e7b0d7d1cb3d4b1a88bf&pid=1-s2.0-S0195669824000891-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141396945","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":"t-sails and sparse hereditary classes of unbounded tree-width","authors":"D. Cocks","doi":"10.1016/j.ejc.2024.104005","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.104005","url":null,"abstract":"<div><p>It has long been known that the following basic objects are obstructions to bounded tree-width: for arbitrarily large <span><math><mi>t</mi></math></span>, <span><math><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></math></span> the complete graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span>, <span><math><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></math></span> the complete bipartite graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>t</mi></mrow></msub></math></span>, <span><math><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></math></span> a subdivision of the <span><math><mrow><mo>(</mo><mi>t</mi><mo>×</mo><mi>t</mi><mo>)</mo></mrow></math></span>-wall and <span><math><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow></math></span> the line graph of a subdivision of the <span><math><mrow><mo>(</mo><mi>t</mi><mo>×</mo><mi>t</mi><mo>)</mo></mrow></math></span>-wall. We now add a further <em>boundary object</em> to this list, a <span><math><mi>t</mi></math></span>-<em>sail</em>. These results have been obtained by studying sparse hereditary <em>path-star</em> graph classes, each of which consists of the finite induced subgraphs of a single infinite graph whose edges can be partitioned into a path (or forest of paths) with a forest of stars, characterised by an infinite word over a possibly infinite alphabet. We show that a path-star class whose infinite graph has an unbounded number of stars, each of which connects an unbounded number of times to the path, has unbounded tree-width. In addition, we show that such a class is not a subclass of the hereditary class of circle graphs. We identify a collection of <em>nested</em> words with a recursive structure that exhibit interesting characteristics when used to define a path-star graph class. These graph classes do not contain any of the four basic obstructions but instead contain graphs that have large tree-width if and only if they contain arbitrarily large <span><math><mi>t</mi></math></span>-sails. We show that these classes are infinitely defined and, like classes of bounded degree or classes excluding a fixed minor, do not contain a minimal class of unbounded tree-width.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"122 ","pages":"Article 104005"},"PeriodicalIF":1.0,"publicationDate":"2024-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824000908/pdfft?md5=e4d9091488efe1ad037850e52d6372a3&pid=1-s2.0-S0195669824000908-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141298105","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":"Special Issue dedicated to the memory of Pierre Rosenstiehl","authors":"Robert Cori (Editors),&nbsp;Jaroslav Nešetřil,&nbsp;Patrice Ossona de Mendez","doi":"10.1016/j.ejc.2023.103800","DOIUrl":"10.1016/j.ejc.2023.103800","url":null,"abstract":"","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"119 ","pages":"Article 103800"},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135388042","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":"Permutation Tutte polynomial","authors":"Csongor Beke ,&nbsp;Gergely Kál Csáji ,&nbsp;Péter Csikvári ,&nbsp;Sára Pituk","doi":"10.1016/j.ejc.2024.104003","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.104003","url":null,"abstract":"<div><p>The classical Tutte polynomial is a two-variate polynomial <span><math><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></math></span> associated to graphs or more generally, matroids. In this paper, we introduce a polynomial <span><math><mrow><msub><mrow><mover><mrow><mi>T</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>H</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></math></span> associated to a bipartite graph <span><math><mi>H</mi></math></span> that we call the permutation Tutte polynomial of the graph <span><math><mi>H</mi></math></span>. It turns out that <span><math><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msub><mrow><mover><mrow><mi>T</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>H</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></math></span> share many properties, and the permutation Tutte polynomial serves as a tool to study the classical Tutte polynomial. We discuss the analogues of Brylawsi’s identities and Conde–Merino–Welsh type inequalities. In particular, we will show that if <span><math><mi>H</mi></math></span> does not contain isolated vertices, then <span><span><span><math><mrow><msub><mrow><mover><mrow><mi>T</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>H</mi></mrow></msub><mrow><mo>(</mo><mn>3</mn><mo>,</mo><mn>0</mn><mo>)</mo></mrow><msub><mrow><mover><mrow><mi>T</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>H</mi></mrow></msub><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>3</mn><mo>)</mo></mrow><mo>≥</mo><msub><mrow><mover><mrow><mi>T</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>H</mi></mrow></msub><msup><mrow><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo></mrow></math></span></span></span>which gives a short proof of the analogous result of Jackson: <span><span><span><math><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mn>3</mn><mo>,</mo><mn>0</mn><mo>)</mo></mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>3</mn><mo>)</mo></mrow><mo>≥</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>G</mi></mrow></msub><msup><mrow><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span></span></span>\u0000for graphs without loops and bridges. We also give improvement on the constant 3 in this statement by showing that one can replace it with 2.9243.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"120 ","pages":"Article 104003"},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141242539","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":"Wachs permutations, Bruhat order and weak order","authors":"Francesco Brenti ,&nbsp;Paolo Sentinelli","doi":"10.1016/j.ejc.2023.103804","DOIUrl":"10.1016/j.ejc.2023.103804","url":null,"abstract":"<div><p>We study the partial orders<span><span> induced on Wachs and signed Wachs permutations by the Bruhat and </span>weak orders<span> of the symmetric and hyperoctahedral groups. We show that these orders are graded, determine their rank function, characterize their ordering and covering relations, and compute their characteristic polynomials, when partially ordered by Bruhat order, and determine their structure explicitly when partially ordered by right weak order.</span></span></p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"119 ","pages":"Article 103804"},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135849064","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":"Unambiguously coded shifts","authors":"Marie-Pierre Béal ,&nbsp;Dominique Perrin ,&nbsp;Antonio Restivo","doi":"10.1016/j.ejc.2023.103812","DOIUrl":"10.1016/j.ejc.2023.103812","url":null,"abstract":"<div><p>We study the coded shifts introduced by Blanchard and Hansel (1986). We give several constructions which allow one to represent a coded shift as an unambiguous one.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"119 ","pages":"Article 103812"},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135894828","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":"Theorems and conjectures on some rational generating functions","authors":"Richard P. Stanley","doi":"10.1016/j.ejc.2023.103814","DOIUrl":"10.1016/j.ejc.2023.103814","url":null,"abstract":"<div><p>Let <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> denote the <span><math><mi>i</mi></math></span>th Fibonacci number, and define <span><math><mrow><msubsup><mrow><mo>∏</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><mfenced><mrow><mn>1</mn><mo>+</mo></mrow></mfenced><mfenced><mrow><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msup></mrow></mfenced><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></math></span>. The paper is concerned primarily with the coefficients <span><math><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow></math></span>. In particular, for any <span><math><mrow><mi>r</mi><mo>≥</mo><mn>0</mn></mrow></math></span> the generating function <span><math><mrow><msub><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub><mrow><mo>(</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo></mrow><msup><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span> is rational. The coefficients <span><math><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow></math></span> can be displayed in an array called the <span><em>Fibonacci triangle </em><em>poset</em></span> <span><math><mi>F</mi></math></span><span> with some interesting further properties, including an encoding of a certain dense linear order on the nonnegative integers. Some generalizations are briefly considered, but there remain many open questions.</span></p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"119 ","pages":"Article 103814"},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135685696","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":"Guarding isometric subgraphs and cops and robber in planar graphs","authors":"Sebastián González Hermosillo de la Maza,&nbsp;Bojan Mohar","doi":"10.1016/j.ejc.2023.103809","DOIUrl":"10.1016/j.ejc.2023.103809","url":null,"abstract":"<div><p>In the game of Cops and Robbers, one of the most useful results is that an isometric path in a graph can be guarded by one cop. In this paper, we introduce the concept of wide shadow in a subgraph, and use it to characterize all 1-guardable graphs. As an application, we show that 3 cops can capture a robber in any planar graph with the added restriction that at most two cops can move simultaneously, proving a conjecture of Yang and strengthening a classical result of Aigner and Fromme.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"119 ","pages":"Article 103809"},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135349188","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":"Kempe changes in degenerate graphs","authors":"Marthe Bonamy,&nbsp;Vincent Delecroix,&nbsp;Clément Legrand–Duchesne","doi":"10.1016/j.ejc.2023.103802","DOIUrl":"10.1016/j.ejc.2023.103802","url":null,"abstract":"<div><p>We consider Kempe changes on the <span><math><mi>k</mi></math></span>-colorings of a graph on <span><math><mi>n</mi></math></span> vertices. If the graph is <span><math><mrow><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>-degenerate, then all its <span><math><mi>k</mi></math></span>-colorings are equivalent up to Kempe changes. However, the sequence between two <span><math><mi>k</mi></math></span>-colorings that arises from the proof may have length exponential in the number of vertices. An intriguing open question is whether it can be turned polynomial. We prove this to be possible under the stronger assumption that the graph has treewidth at most <span><math><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></math></span>. Namely, any two <span><math><mi>k</mi></math></span>-colorings are equivalent up to <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>k</mi><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> Kempe changes. We investigate other restrictions (list coloring, bounded maximum average degree, degree bounds). As one of the main results, we derive that given an <span><math><mi>n</mi></math></span><span>-vertex graph with maximum degree </span><span><math><mi>Δ</mi></math></span>, the <span><math><mi>Δ</mi></math></span>-colorings are all equivalent up to <span><math><mrow><msub><mrow><mi>O</mi></mrow><mrow><mi>Δ</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> Kempe changes, unless <span><math><mrow><mi>Δ</mi><mo>=</mo><mn>3</mn></mrow></math></span> and some connected component is a 3-prism, that is <span><math><mrow><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>□</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math></span>, in which case there exist some non-equivalent 3-colorings.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"119 ","pages":"Article 103802"},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138536325","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":"Minimum lethal sets in grids and tori under 3-neighbour bootstrap percolation","authors":"Fabricio Benevides ,&nbsp;Jean-Claude Bermond ,&nbsp;Hicham Lesfari ,&nbsp;Nicolas Nisse","doi":"10.1016/j.ejc.2023.103801","DOIUrl":"10.1016/j.ejc.2023.103801","url":null,"abstract":"<div><p>Let <span><math><mrow><mi>r</mi><mo>≥</mo><mn>1</mn></mrow></math></span><span> be any non negative integer and let </span><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> be any undirected graph in which a subset <span><math><mrow><mi>D</mi><mo>⊆</mo><mi>V</mi></mrow></math></span> of vertices are initially <em>infected</em>. We consider the process in which, at every step, each non-infected vertex with at least <span><math><mi>r</mi></math></span> infected neighbours becomes infected and an infected vertex never becomes non-infected. The problem consists in determining the minimum size <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> of an initially infected vertices set <span><math><mi>D</mi></math></span> that eventually infects the whole graph <span><math><mi>G</mi></math></span>. This problem is closely related to cellular automata, to percolation problems and to the Game of Life studied by John Conway. Note that <span><math><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span><span> for any connected graph </span><span><math><mi>G</mi></math></span>. The case when <span><math><mi>G</mi></math></span> is the <span><math><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></math></span> grid, <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub></math></span>, and <span><math><mrow><mi>r</mi><mo>=</mo><mn>2</mn></mrow></math></span> is well known and appears in many puzzle books, in particular due to the elegant proof that shows that <span><math><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mi>n</mi></mrow></math></span> for all <span><math><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></math></span>. We study the cases of square grids, <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub></math></span>, and tori, <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub></math></span>, when <span><math><mrow><mi>r</mi><mo>∈</mo><mrow><mo>{</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>}</mo></mrow></mrow></math></span>. We show that <span><math><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>3</mn></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mrow><mo>⌈</mo><mfrac><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>4</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>⌉</mo></mrow></mrow></math></span> for every <span><math><mi>n</mi></math></span> even and that <span><math><mrow><mrow><mo>⌈</mo><mfrac><mrow><ms","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"119 ","pages":"Article 103801"},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135894821","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}