Discrete Mathematics最新文献

{"title":"Disconnected forbidden pairs force supereulerian graphs to be hamiltonian","authors":"Qiang Wang ,&nbsp;Liming Xiong","doi":"10.1016/j.disc.2024.114301","DOIUrl":"10.1016/j.disc.2024.114301","url":null,"abstract":"<div><div>A graph is said to be supereulerian if it has a spanning eulerian subgraph, i.e., a spanning connected even subgraph. A graph is called hamiltonian if it contains a spanning cycle. A graph is said to be <span><math><mo>{</mo><mi>R</mi><mo>,</mo><mi>S</mi><mo>}</mo></math></span>-free if it does not contain <em>R</em> or <em>S</em> as an induced subgraph. Yang et al. characterized all pairs of connected graphs <span><math><mi>R</mi><mo>,</mo><mi>S</mi></math></span> such that every supereulerian <span><math><mo>{</mo><mi>R</mi><mo>,</mo><mi>S</mi><mo>}</mo></math></span>-free graph is hamiltonian. In this paper, we consider disconnected forbidden graph <span><math><mi>R</mi><mo>,</mo><mi>S</mi></math></span>. We characterize all pairs of disconnected graphs <span><math><mi>R</mi><mo>,</mo><mi>S</mi></math></span> such that every supereulerian <span><math><mo>{</mo><mi>R</mi><mo>,</mo><mi>S</mi><mo>}</mo></math></span>-free graph of sufficiently large order is hamiltonian. Applying this result, we also characterize all forbidden pairs for the existence of a Hamiltonian cycle in 2-edge connected graphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114301"},"PeriodicalIF":0.7,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142560886","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":"Rigid frameworks with dilation constraints","authors":"Sean Dewar ,&nbsp;Anthony Nixon ,&nbsp;Andrew Sainsbury","doi":"10.1016/j.disc.2024.114304","DOIUrl":"10.1016/j.disc.2024.114304","url":null,"abstract":"<div><div>We consider the rigidity and global rigidity of bar-joint frameworks in Euclidean <em>d</em>-space under additional dilation constraints in specified coordinate directions. In this setting we obtain a complete characterisation of generic rigidity. We then consider generic global rigidity. In particular, we provide an algebraic sufficient condition and a weak necessary condition. We also construct a large family of globally rigid frameworks and conjecture a combinatorial characterisation when most coordinate directions have dilation constraints.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114304"},"PeriodicalIF":0.7,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142560888","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":"A note on locating-dominating sets in twin-free graphs","authors":"Nicolas Bousquet ,&nbsp;Quentin Chuet ,&nbsp;Victor Falgas–Ravry ,&nbsp;Amaury Jacques ,&nbsp;Laure Morelle","doi":"10.1016/j.disc.2024.114297","DOIUrl":"10.1016/j.disc.2024.114297","url":null,"abstract":"<div><div>In this short note, we prove that every twin-free graph on <em>n</em> vertices contains a locating-dominating set of size at most <span><math><mo>⌈</mo><mfrac><mrow><mn>5</mn></mrow><mrow><mn>8</mn></mrow></mfrac><mi>n</mi><mo>⌉</mo></math></span>. This improves the earlier bound of <span><math><mo>⌊</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mi>n</mi><mo>⌋</mo></math></span> due to Foucaud, Henning, Löwenstein and Sasse from 2016, and makes some progress towards the well-studied locating-dominating conjecture of Garijo, González and Márquez.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114297"},"PeriodicalIF":0.7,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554469","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":"Generation of 3-connected, planar line graphs","authors":"Phoebe Hollowbread-Smith ,&nbsp;Riccardo W. Maffucci","doi":"10.1016/j.disc.2024.114302","DOIUrl":"10.1016/j.disc.2024.114302","url":null,"abstract":"<div><div>We classify and construct all line graphs that are 3-polytopes (planar and 3-connected). Apart from a few special cases, they are all obtained starting from the medial graphs of cubic (i.e., 3-regular) 3-polytopes, by applying two types of graph transformations. This is similar to the generation of other subclasses of 3-polytopes <span><span>[6]</span></span>, <span><span>[13]</span></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114302"},"PeriodicalIF":0.7,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554454","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":"Light 3-faces in 3-polytopes without adjacent triangles","authors":"O.V. Borodin ,&nbsp;A.O. Ivanova","doi":"10.1016/j.disc.2024.114299","DOIUrl":"10.1016/j.disc.2024.114299","url":null,"abstract":"<div><div>Over the last decades, a lot of research has been devoted to structural and coloring problems on plane graphs that are sparse in this or that sense.</div><div>In this note we deal with the densest among sparse 3-polytopes, namely those having no adjacent 3-cycles. Borodin (1996) proved that such 3-polytopes have a vertex of degree at most 4 and, moreover, an edge with the degree-sum of its end-vertices at most 9, where both bounds are sharp.</div><div>By <span><math><mi>d</mi><mo>(</mo><mi>v</mi><mo>)</mo></math></span> denote the degree of a vertex <em>v</em>. An edge <span><math><mi>e</mi><mo>=</mo><mi>x</mi><mi>y</mi></math></span> in a 3-polytope is an <span><math><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>)</mo></math></span>-edge if <span><math><mi>d</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>≤</mo><mi>i</mi></math></span> and <span><math><mi>d</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>≤</mo><mi>j</mi></math></span>. The well-known (3,5;4,4)-Archimedean solid corresponds to a plane quadrangulation in which every edge joins a 3-vertex with a 5-vertex.</div><div>We prove that every 3-polytope with neither adjacent 3-cycles nor <span><math><mo>(</mo><mn>3</mn><mo>,</mo><mn>5</mn><mo>)</mo></math></span>-edges has a 3-face with the degree-sum of its incident vertices (weight) at most 16, which bound is sharp.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 1","pages":"Article 114299"},"PeriodicalIF":0.7,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142529640","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":"Counting spanning trees of multiple complete split-like graph containing a given spanning forest","authors":"Chenlin Yang,&nbsp;Tao Tian","doi":"10.1016/j.disc.2024.114300","DOIUrl":"10.1016/j.disc.2024.114300","url":null,"abstract":"<div><div>The multiple complete split-like graph <span><math><mi>M</mi><mi>C</mi><msubsup><mrow><mi>S</mi></mrow><mrow><mi>b</mi><mo>,</mo><mi>s</mi></mrow><mrow><mi>a</mi></mrow></msubsup></math></span> is the join of an empty graph <span><math><msub><mrow><mover><mrow><mi>K</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>a</mi></mrow></msub></math></span> and <em>s</em> copies of complete graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>b</mi></mrow></msub></math></span>. In this article, we obtain the formulas for the number of spanning trees of <span><math><mi>M</mi><mi>C</mi><msubsup><mrow><mi>S</mi></mrow><mrow><mi>b</mi><mo>,</mo><mi>s</mi></mrow><mrow><mi>a</mi></mrow></msubsup></math></span> containing a given spanning forest when <span><math><mi>s</mi><mo>=</mo><mn>1</mn></math></span> and 2. Particularly, when <span><math><mi>s</mi><mo>=</mo><mn>1</mn></math></span>, our result derives the number of spanning trees of complete split graph containing a given spanning forest, thereby extending Moon's result <span><span>[19]</span></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114300"},"PeriodicalIF":0.7,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142534362","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":"Large matchings in maximal 1-planar graphs","authors":"Therese Biedl ,&nbsp;John Wittnebel","doi":"10.1016/j.disc.2024.114288","DOIUrl":"10.1016/j.disc.2024.114288","url":null,"abstract":"<div><div>It is well-known that every maximal planar graph has a matching of size at least <span><math><mfrac><mrow><mi>n</mi><mo>+</mo><mn>8</mn></mrow><mrow><mn>3</mn></mrow></mfrac></math></span> if <span><math><mi>n</mi><mo>≥</mo><mn>14</mn></math></span>. In this paper, we investigate similar matching-bounds for maximal <em>1-planar</em> graphs, i.e., graphs that can be drawn such that every edge has at most one crossing. In particular we show that every 3-connected simple-maximal 1-planar graph has a matching of size at least <span><math><mfrac><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>6</mn></mrow><mrow><mn>5</mn></mrow></mfrac></math></span>; the bound decreases to <span><math><mfrac><mrow><mn>3</mn><mi>n</mi><mo>+</mo><mn>14</mn></mrow><mrow><mn>10</mn></mrow></mfrac></math></span> if the graph need not be 3-connected. We also give (weaker) bounds when the graph comes with a fixed 1-planar drawing or is not simple. All our bounds are tight in the sense that some graph that satisfies the restrictions has no bigger matching.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114288"},"PeriodicalIF":0.7,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142534361","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":"The decycling number of a line graph","authors":"Mingyuan Ma,&nbsp;Han Ren","doi":"10.1016/j.disc.2024.114291","DOIUrl":"10.1016/j.disc.2024.114291","url":null,"abstract":"<div><div>The decycling number of a graph <em>G</em>, denoted by <span><math><mi>∇</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, is the number of vertices in a minimum decycling set of <em>G</em>. The line graph of <em>G</em> is denoted by <span><math><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. In this paper we show that <span><math><mi>∇</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>=</mo><mi>β</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mi>μ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span>, where <span><math><mi>β</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is the cycle rank of <em>G</em> and <span><math><mi>μ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is the path partition number of <em>G</em>. In particular, <span><math><mi>∇</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>=</mo><mi>β</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> if and only if <em>G</em> has a Hamilton path, and <span><math><mi>∇</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>≤</mo><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mrow><mn>3</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac></math></span> if <em>G</em> is a cubic graph with <em>n</em> vertices, where <span><math><mi>n</mi><mo>≥</mo><mn>10</mn></math></span>. If <span><math><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is a planar graph, then we prove that <span><math><mi>∇</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>≤</mo><mfrac><mrow><mo>|</mo><mi>V</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>|</mo></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>, which means that the conjecture proposed by Albertson and Berman in 1979 that the decycling number of any planar graph <em>H</em> is at most <span><math><mfrac><mrow><mo>|</mo><mi>V</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>|</mo></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> holds for a planar line graph. If <em>G</em> is a connected graph of order <em>n</em> which is 2-cell embedded on the orientable surface <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>g</mi></mrow></msub></math></span> (or the non-orientable surface <span><math><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>k</mi></mrow><mrow><mo>′</mo></mrow></msubsup></math></span>), then we show that <span><math><mi>∇</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>≤</mo><mn>2</mn><mi>n</mi><mo>+</mo><mi>l</mi><mo>−</mo><mn>7</mn><mo>+</mo><mn>6</mn><mi>g</mi></math></span> (or <span><math><mn>2</mn><mi>n</mi><mo>+</mo><mi>l</mi><mo>−</mo><mn>7</mn><mo>+</mo><mn>3</mn><mi>k</mi></math></span>) if <em>G</em> has a spanning tree with <em>l</em> leaves. Our bounds are tight for <span><math><mi>l</mi><mo>=</mo><mn>2</mn></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114291"},"PeriodicalIF":0.7,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142534363","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 transient equivalence between Aldous-Broder and Wilson's algorithms and a two-stage framework for generating uniform spanning trees","authors":"Igor Nunes ,&nbsp;Giulio Iacobelli ,&nbsp;Daniel Ratton Figueiredo","doi":"10.1016/j.disc.2024.114285","DOIUrl":"10.1016/j.disc.2024.114285","url":null,"abstract":"<div><div>The <em>Aldous-Broder</em> and <em>Wilson</em> are two well-known algorithms for generating uniform spanning trees (USTs) based on random walks. This work studies their transient relationship by introducing the notion of <em>branches</em>—paths generated by the two algorithms on particular stopping times, in order to show that the trees built by the two algorithms when running on a complete graph are statistically equivalent on these stopping times. This leads to a hybrid algorithm that can generate USTs faster than either of the two algorithms. The idea is generalized to a two-stage framework to generate USTs on arbitrary graphs. The feasibility of the framework is shown through various examples, including some edge transitive graphs where the average running time can be 25% smaller than <em>Wilson</em> to generate USTs. Results obtained through numerical simulations of the framework on complete graphs and hypercubes illustrate the findings.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114285"},"PeriodicalIF":0.7,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142534360","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":"2-Distance (Δ + 1)-coloring of sparse graphs using the potential method","authors":"Hoang La,&nbsp;Mickael Montassier","doi":"10.1016/j.disc.2024.114292","DOIUrl":"10.1016/j.disc.2024.114292","url":null,"abstract":"<div><div>A 2-distance <em>k</em>-coloring of a graph is a proper <em>k</em>-coloring of the vertices where vertices at distance at most 2 cannot share the same color. We prove the existence of a 2-distance (<span><math><mi>Δ</mi><mo>+</mo><mn>1</mn></math></span>)-coloring for graphs with maximum average degree less than <span><math><mfrac><mrow><mn>18</mn></mrow><mrow><mn>7</mn></mrow></mfrac></math></span> and maximum degree <span><math><mi>Δ</mi><mo>≥</mo><mn>7</mn></math></span>. As a corollary, every planar graph with girth at least 9 and <span><math><mi>Δ</mi><mo>≥</mo><mn>7</mn></math></span> admits a 2-distance <span><math><mo>(</mo><mi>Δ</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-coloring. The proof uses the potential method to reduce new configurations compared to classic approaches on 2-distance coloring.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 1","pages":"Article 114292"},"PeriodicalIF":0.7,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142529717","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}