{"title":"Light 3-faces in 3-polytopes without adjacent triangles","authors":"O.V. Borodin , 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, 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 , 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, 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}
Igor Nunes , Giulio Iacobelli , Daniel Ratton Figueiredo
{"title":"A transient equivalence between Aldous-Broder and Wilson's algorithms and a two-stage framework for generating uniform spanning trees","authors":"Igor Nunes , Giulio Iacobelli , 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, 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}
{"title":"On the size and structure of t-representable sumsets","authors":"Christian Táfula","doi":"10.1016/j.disc.2024.114295","DOIUrl":"10.1016/j.disc.2024.114295","url":null,"abstract":"<div><div>Let <span><math><mi>A</mi><mo>⊆</mo><msub><mrow><mi>Z</mi></mrow><mrow><mo>≥</mo><mn>0</mn></mrow></msub></math></span> be a finite set with minimum element 0, maximum element <em>m</em>, and <em>ℓ</em> elements strictly in between. Write <span><math><msup><mrow><mo>(</mo><mi>h</mi><mi>A</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></msup></math></span> for the set of integers that can be written in at least <em>t</em> ways as a sum of <em>h</em> elements of <em>A</em>. We prove that <span><math><msup><mrow><mo>(</mo><mi>h</mi><mi>A</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></msup></math></span> is “structured” for<span><span><span><math><mi>h</mi><mo>≥</mo><mo>(</mo><mn>1</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>e</mi></mrow></mfrac><mi>m</mi><mi>ℓ</mi><msup><mrow><mi>t</mi></mrow><mrow><mn>1</mn><mo>/</mo><mi>ℓ</mi></mrow></msup></math></span></span></span> (as <span><math><mi>ℓ</mi><mo>→</mo><mo>∞</mo></math></span>, <span><math><msup><mrow><mi>t</mi></mrow><mrow><mn>1</mn><mo>/</mo><mi>ℓ</mi></mrow></msup><mo>→</mo><mo>∞</mo></math></span>), and prove a similar theorem on the size and structure of <span><math><mi>A</mi><mo>⊆</mo><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> for <em>h</em> sufficiently large. Moreover, we construct a family of sets <span><math><mi>A</mi><mo>=</mo><mi>A</mi><mo>(</mo><mi>m</mi><mo>,</mo><mi>ℓ</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>⊆</mo><msub><mrow><mi>Z</mi></mrow><mrow><mo>≥</mo><mn>0</mn></mrow></msub></math></span> for which <span><math><msup><mrow><mo>(</mo><mi>h</mi><mi>A</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></msup></math></span> is not structured for <span><math><mi>h</mi><mo>≪</mo><mi>m</mi><mi>ℓ</mi><msup><mrow><mi>t</mi></mrow><mrow><mn>1</mn><mo>/</mo><mi>ℓ</mi></mrow></msup></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 1","pages":"Article 114295"},"PeriodicalIF":0.7,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142529639","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":"On the number of small Steiner triple systems with Veblen points","authors":"Giuseppe Filippone , Mario Galici","doi":"10.1016/j.disc.2024.114294","DOIUrl":"10.1016/j.disc.2024.114294","url":null,"abstract":"<div><div>The concept of <em>Schreier extensions</em> of loops was introduced in the general case in <span><span>[11]</span></span> and, more recently, it has been explored in the context of Steiner loops in <span><span>[6]</span></span>. In the latter case, it gives a powerful method for constructing Steiner triple systems containing Veblen points. Counting all Steiner triple systems of order <em>v</em> is an open problem for <span><math><mi>v</mi><mo>></mo><mn>21</mn></math></span>. In this paper, we investigate the number of Steiner triple systems of order 19, 27 and 31 containing Veblen points and we present some examples.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 1","pages":"Article 114294"},"PeriodicalIF":0.7,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142529718","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 average connectivity matrix of a graph","authors":"Linh Nguyen , Suil O","doi":"10.1016/j.disc.2024.114290","DOIUrl":"10.1016/j.disc.2024.114290","url":null,"abstract":"<div><div>For a graph <em>G</em> and for two distinct vertices <em>u</em> and <em>v</em>, let <span><math><mi>κ</mi><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></math></span> be the maximum number of vertex-disjoint paths joining <em>u</em> and <em>v</em> in <em>G</em>. The average connectivity matrix of an <em>n</em>-vertex connected graph <em>G</em>, written <span><math><msub><mrow><mi>A</mi></mrow><mrow><mover><mrow><mi>κ</mi></mrow><mo>‾</mo></mover></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, is an <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrix whose <span><math><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></math></span>-entry is <span><math><mi>κ</mi><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo><mo>/</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow></math></span> and let <span><math><mi>ρ</mi><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mover><mrow><mi>κ</mi></mrow><mo>‾</mo></mover></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span> be the spectral radius of <span><math><msub><mrow><mi>A</mi></mrow><mrow><mover><mrow><mi>κ</mi></mrow><mo>‾</mo></mover></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. In this paper, we investigate some spectral properties of the matrix. In particular, we prove that for any <em>n</em>-vertex connected graph <em>G</em>, we have <span><math><mi>ρ</mi><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mover><mrow><mi>κ</mi></mrow><mo>‾</mo></mover></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>≤</mo><mfrac><mrow><mn>4</mn><msup><mrow><mi>α</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mrow><mi>n</mi></mrow></mfrac></math></span>, which implies a result of Kim and O <span><span>[8]</span></span> stating that for any connected graph <em>G</em>, we have <span><math><mover><mrow><mi>κ</mi></mrow><mo>‾</mo></mover><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mn>2</mn><msup><mrow><mi>α</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, where <span><math><mover><mrow><mi>κ</mi></mrow><mo>‾</mo></mover><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>u</mi><mo>,</mo><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub><mfrac><mrow><mi>κ</mi><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow></mfrac></math></span> and <span><math><msup><mrow><mi>α</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is the maximum size of a matching in <em>G</em>; equality holds only when <em>G</em> is a complete graph with an odd number of vertices. Also, for bipartite graphs, we improve the bound, namely <span><math><mi>ρ</mi><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mover><mrow><mi>κ</mi></mrow><mo>‾</mo></mover></mro","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114290"},"PeriodicalIF":0.7,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142534359","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":"Equitable coloring in 1-planar graphs","authors":"Daniel W. Cranston, Reem Mahmoud","doi":"10.1016/j.disc.2024.114286","DOIUrl":"10.1016/j.disc.2024.114286","url":null,"abstract":"<div><div>For every <span><math><mi>r</mi><mo>≥</mo><mn>13</mn></math></span>, we show every 1-planar graph <em>G</em> with <span><math><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>r</mi></math></span> has an equitable <em>r</em>-coloring.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114286"},"PeriodicalIF":0.7,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142534357","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}
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