European Journal of Combinatorics最新文献

{"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}

{"title":"Walks avoiding a quadrant and the reflection principle","authors":"Mireille Bousquet-Mélou,&nbsp;Michael Wallner","doi":"10.1016/j.ejc.2023.103803","DOIUrl":"10.1016/j.ejc.2023.103803","url":null,"abstract":"<div><p>We continue the enumeration of plane lattice walks with small steps avoiding the negative quadrant, initiated by the first author in 2016. We solve in detail a new case, namely the king model where all eight nearest neighbour steps are allowed. The associated generating function is proved to be the sum of a simple, explicit D-finite series (related to the number of walks confined to the first quadrant), and an algebraic one. This was already the case for the two models solved by the first author in 2016. The principle of the approach is also the same, but challenging theoretical and computational difficulties arise as we now handle algebraic series of larger degree.</p><p>We expect a similar algebraicity phenomenon to hold for the seven <em>Weyl</em> step sets, which are those for which walks confined to the first quadrant can be counted using the reflection principle. With this paper, this is now proved for three of them. For the remaining four, we predict the D-finite part of the solution, and in three of the four cases, give evidence for the algebraicity of the remaining part.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136119309","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":"Group coloring and group connectivity with non-isomorphic groups of the same order","authors":"Rikke Langhede,&nbsp;Carsten Thomassen","doi":"10.1016/j.ejc.2023.103816","DOIUrl":"10.1016/j.ejc.2023.103816","url":null,"abstract":"<div><p>For every natural number <span><math><mi>k</mi></math></span>, there exists a planar graph which is <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msubsup></math></span>-colorable, but not <span><math><mi>Γ</mi></math></span>-colorable for any other Abelian group <span><math><mi>Γ</mi></math></span> of order <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msup></math></span>. Its dual graph is <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msubsup></math></span>-connected, but not <span><math><mi>Γ</mi></math></span>-connected for any other Abelian group <span><math><mi>Γ</mi></math></span> of order <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msup></math></span>.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669823001336/pdfft?md5=4d2c19bd40b56019f9f9869e2dedb235&pid=1-s2.0-S0195669823001336-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135348627","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":"Extremal graphs without long paths and large cliques","authors":"Gyula O.H. Katona ,&nbsp;Chuanqi Xiao","doi":"10.1016/j.ejc.2023.103807","DOIUrl":"10.1016/j.ejc.2023.103807","url":null,"abstract":"<div><p>Let <span><math><mi>F</mi></math></span> be a family of graphs. A graph is called <span><math><mi>F</mi></math></span>-free if it does not contain any member of <span><math><mi>F</mi></math></span> as a subgraph. The Turán number of <span><math><mi>F</mi></math></span> is the maximum number of edges in an <span><math><mi>n</mi></math></span>-vertex <span><math><mi>F</mi></math></span>-free graph and is denoted by <span><math><mrow><mo>ex</mo><mrow><mo>(</mo><mrow><mi>n</mi><mo>,</mo><mi>F</mi></mrow><mo>)</mo></mrow></mrow></math></span>. The same maximum under the additional condition that the graphs are connected is <span><math><mrow><msub><mrow><mo>ex</mo></mrow><mrow><mi>conn</mi></mrow></msub><mrow><mo>(</mo><mrow><mi>n</mi><mo>,</mo><mi>F</mi></mrow><mo>)</mo></mrow></mrow></math></span>. Let <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> be the path on <span><math><mi>k</mi></math></span> vertices, <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> be the clique on <span><math><mi>m</mi></math></span> vertices. We determine <span><math><mrow><mo>ex</mo><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mrow><mo>{</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>}</mo></mrow><mo>)</mo></mrow></mrow></math></span> if <span><math><mrow><mi>k</mi><mo>&gt;</mo><mn>2</mn><mi>m</mi><mo>−</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><msub><mrow><mo>ex</mo></mrow><mrow><mi>conn</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mrow><mo>{</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>}</mo></mrow><mo>)</mo></mrow></mrow></math></span> if <span><math><mrow><mi>k</mi><mo>&gt;</mo><mi>m</mi></mrow></math></span> for sufficiently large <span><math><mi>n</mi></math></span>.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669823001245/pdfft?md5=1271cd195bffe5dd20cfa6c9c7c1cd05&pid=1-s2.0-S0195669823001245-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135388839","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":"Finding strong components using depth-first search","authors":"Robert E. Tarjan ,&nbsp;Uri Zwick","doi":"10.1016/j.ejc.2023.103815","DOIUrl":"10.1016/j.ejc.2023.103815","url":null,"abstract":"<div><p>We survey three algorithms that use depth-first search to find the strong components of a directed graph in linear time: (1) Tarjan’s algorithm; (2) a cycle-finding algorithm; and (3) a bidirectional search algorithm.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135963016","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":"Variations on a tree","authors":"Pascale Kuntz ,&nbsp;Bruno Pinaud","doi":"10.1016/j.ejc.2023.103808","DOIUrl":"10.1016/j.ejc.2023.103808","url":null,"abstract":"<div><p>The family tree is like an inherited “object” that has been passed down through many generations, with many and varied definitions which distort the tree both as a combinatorial object and in its visual representations. Moreover, whether used by amateur genealogists or academic researchers, it is always contextualized by both validated exogenous knowledge and by implicit knowledge. In this paper, we explore introducing certain contextual information that is associated with a locally defined dissimilarity between individuals of the same generation. We propose a new heuristic based on a radial representation of a node-link model which seeks to preserve local proximities in the layout. This heuristic is applied in an original form, which is that of Pierre Rosenstiehl’s “scientific family tree”.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134934597","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}