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The phase transitions of diameters in random axis-parallel hyperrectangle intersection graphs
IF 1
3区 数学
Congsong Zhang , Yong Gao , James Nastos
{"title":"The phase transitions of diameters in random axis-parallel hyperrectangle intersection graphs","authors":"Congsong Zhang , Yong Gao , James Nastos","doi":"10.1016/j.dam.2025.01.044","DOIUrl":"10.1016/j.dam.2025.01.044","url":null,"abstract":"<div><div>We study the behaviours of diameters in two models of random axis-parallel hyperrectangle intersection graphs: <span><math><mrow><mi>G</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>l</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>u</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>l</mi><mo>)</mo></mrow></mrow></math></span>. These two models use axis-parallel <span><math><mi>l</mi></math></span>-dimensional hyperrectangles to represent vertices, and vertices are adjacent if and only if their corresponding axis-parallel hyperrectangles intersect. In the model <span><math><mrow><mi>G</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>l</mi><mo>)</mo></mrow></mrow></math></span>, we distribute <span><math><mi>n</mi></math></span> points within <span><math><msup><mrow><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow><mrow><mi>l</mi></mrow></msup></math></span> uniformly and independently, and each point is the centre of an axis-parallel <span><math><mi>l</mi></math></span>-dimensional hypercube with edge length <span><math><mi>r</mi></math></span>. The model <span><math><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>u</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>l</mi><mo>)</mo></mrow></mrow></math></span>, distributing the centres of <span><math><mi>n</mi></math></span> axis-parallel <span><math><mi>l</mi></math></span>-dimensional hyperrectangles within <span><math><msup><mrow><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow><mrow><mi>l</mi></mrow></msup></math></span> exactly as the model <span><math><mrow><mi>G</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>l</mi><mo>)</mo></mrow></mrow></math></span>, assigns a length from a uniform distribution over <span><math><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>r</mi><mo>]</mo></mrow></math></span> to each edge of the <span><math><mi>n</mi></math></span> axis-parallel <span><math><mi>l</mi></math></span>-dimensional hyperrectangles.</div><div>We prove that in the model <span><math><mrow><mi>G</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>l</mi><mo>)</mo></mrow></mrow></math></span>, there is a phase transition for the event that the diameter is at most <span><math><mrow><mi>d</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> occurring at <span><math><mrow><mi>r</mi><mo>=</mo><mi>d</mi><msup><mrow><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span> if <span><math><mrow><mi>n</mi><mi>⋅</mi><mi>d</mi><msup><mrow><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mi>l</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></msup><mo>≥</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>ϵ</mi></mrow></msup><mo>,</mo></mrow></math></span>\u0000 where <span><math><mrow><mn>0</mn><mo><</mo><mi>ϵ</mi><","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"367 ","pages":"Pages 22-29"},"PeriodicalIF":1.0,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143225980","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}
引用次数: 0
Bounds for boxicity of circular clique graphs and zero-divisor graphs
IF 1
3区 数学
T. Kavaskar
{"title":"Bounds for boxicity of circular clique graphs and zero-divisor graphs","authors":"T. Kavaskar","doi":"10.1016/j.dam.2025.01.038","DOIUrl":"10.1016/j.dam.2025.01.038","url":null,"abstract":"<div><div>Let <span><math><mrow><mi>b</mi><mi>o</mi><mi>x</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> be the boxicity of a graph <span><math><mi>G</mi></math></span>, <span><math><mrow><mi>G</mi><mrow><mo>[</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></mrow></mrow></math></span> be the <span><math><mi>G</mi></math></span>-join graph of <span><math><mi>n</mi></math></span>-pairwise disjoint graphs <span><math><mrow><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span>, <span><math><msubsup><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>d</mi></mrow></msubsup></math></span> be a circular clique graph (where <span><math><mrow><mi>k</mi><mo>≥</mo><mn>2</mn><mi>d</mi></mrow></math></span>) and <span><math><mrow><mi>Γ</mi><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span> be the zero-divisor graph of a commutative ring <span><math><mi>R</mi></math></span> with unity. In this paper, we prove that <span><math><mrow><mi>χ</mi><mrow><mo>(</mo><msubsup><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>d</mi></mrow></msubsup><mo>)</mo></mrow><mo>≥</mo><mi>b</mi><mi>o</mi><mi>x</mi><mrow><mo>(</mo><msubsup><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>d</mi></mrow></msubsup><mo>)</mo></mrow></mrow></math></span>, for all <span><math><mi>k</mi></math></span> and <span><math><mi>d</mi></math></span> with <span><math><mrow><mi>k</mi><mo>≥</mo><mn>2</mn><mi>d</mi></mrow></math></span>. This generalizes the results proved by Kamibeppu (2018). Also we obtain that <span><math><mrow><mi>b</mi><mi>o</mi><mi>x</mi><mrow><mo>(</mo><mi>G</mi><mrow><mo>[</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></mrow><mo>)</mo></mrow><mo>≤</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><mi>b</mi><mi>o</mi><mi>x</mi><mrow><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>. As a consequence of this result, we obtain a bound for boxicity of ideal-based zero-divisor graph of a finite commutative ring with unity. In particular, if <span><math><mrow><mi>R</mi><mo>≇</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>×</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></mrow></math></span> is a finite commutative non-zero reduced ring with unity, where <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></sp","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"365 ","pages":"Pages 260-269"},"PeriodicalIF":1.0,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150139","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}
引用次数: 0
Near optimal colourability on (H, Kn−e)-free graphs
IF 1
3区 数学
Yiao Ju , Shenwei Huang
{"title":"Near optimal colourability on (H, Kn−e)-free graphs","authors":"Yiao Ju , Shenwei Huang","doi":"10.1016/j.dam.2025.01.042","DOIUrl":"10.1016/j.dam.2025.01.042","url":null,"abstract":"<div><div>A graph family <span><math><mi>G</mi></math></span> is <em>near optimal colourable</em> if there is a constant number <span><math><mi>c</mi></math></span>, such that every graph <span><math><mrow><mi>G</mi><mo>∈</mo><mi>G</mi></mrow></math></span> satisfies <span><math><mrow><mi>χ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mo>max</mo><mrow><mo>{</mo><mi>c</mi><mo>,</mo><mi>ω</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>}</mo></mrow></mrow></math></span>, where <span><math><mrow><mi>χ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>ω</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> are the chromatic number and clique number of <span><math><mi>G</mi></math></span>, respectively. One may reduce the colouring problem on a near optimal colourable graph family to <span><math><mi>q</mi></math></span>-colouring problems for <span><math><mrow><mi>q</mi><mo>≤</mo><mi>c</mi><mo>−</mo><mn>1</mn></mrow></math></span>. In our previous paper [Y. Ju and S. Huang. Near optimal colourability on hereditary graph families. Theoretical Computer Science 994: 114465, 2024], we give an almost complete characterization for the near optimal colourability for (<span><math><mrow><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>)-free graphs and give the open problem: “Decide whether the family of (<span><math><mrow><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>)-free graphs is near optimal colourable, when <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> is a forest with independence number at least 3 and <span><math><mrow><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>−</mo><mi>e</mi></mrow></math></span>\u0000 (<span><math><mrow><mi>n</mi><mo>≥</mo><mn>4</mn></mrow></math></span>).” In this paper, we partially solve this open problem. We prove that for every <span><math><mrow><mi>n</mi><mo>≥</mo><mn>4</mn></mrow></math></span>, the family of (<span><math><mi>H</mi></math></span>, <span><math><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>−</mo><mi>e</mi></mrow></math></span>)-free graphs is near optimal colourable if <span><math><mi>H</mi></math></span> is an induced subgraph of <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span>, <span><math><mrow><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>+</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span> or <span><math><mrow><mi>m</mi><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span> for any <span><math><mi>m</mi></ma","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"367 ","pages":"Pages 1-7"},"PeriodicalIF":1.0,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143168316","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}
引用次数: 0
Counting r×s rectangles in (Catalan) words
IF 1
3区 数学
Sela Fried , Toufik Mansour
{"title":"Counting r×s rectangles in (Catalan) words","authors":"Sela Fried , Toufik Mansour","doi":"10.1016/j.dam.2025.01.031","DOIUrl":"10.1016/j.dam.2025.01.031","url":null,"abstract":"<div><div>Generalizing previous results, we introduce and study a new statistic on words, that we call rectangle capacity. For two fixed positive integers <span><math><mi>r</mi></math></span> and <span><math><mi>s</mi></math></span>, this statistic counts the number of occurrences of a rectangle of size <span><math><mrow><mi>r</mi><mo>×</mo><mi>s</mi></mrow></math></span> in the bargraph representation of a word. We find the bivariate generating function for the distribution on words of the number of <span><math><mrow><mi>r</mi><mo>×</mo><mi>s</mi></mrow></math></span> rectangles and the generating function for their total number over all words. We also obtain the analog results for Catalan words.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"365 ","pages":"Pages 247-259"},"PeriodicalIF":1.0,"publicationDate":"2025-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150625","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}
引用次数: 0
VC-dimension and pseudo-random graphs
IF 1
3区 数学
Thang Pham , Steven Senger , Michael Tait , Nguyen Thu-Huyen
{"title":"VC-dimension and pseudo-random graphs","authors":"Thang Pham , Steven Senger , Michael Tait , Nguyen Thu-Huyen","doi":"10.1016/j.dam.2025.01.030","DOIUrl":"10.1016/j.dam.2025.01.030","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a graph and <span><math><mrow><mi>U</mi><mo>⊂</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> be a set of vertices. For each <span><math><mrow><mi>v</mi><mo>∈</mo><mi>U</mi></mrow></math></span>, let <span><math><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>:</mo><mi>U</mi><mo>→</mo><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></mrow></math></span> be the function defined by <span><span><span><math><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>v</mi></mrow></msub><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mfenced><mrow><mtable><mtr><mtd><mi>&</mi><mn>1</mn><mspace></mspace><mtext>if</mtext><mspace></mspace><mi>u</mi><mo>∼</mo><mi>v</mi><mo>,</mo><mi>u</mi><mo>∈</mo><mi>U</mi><mspace></mspace></mtd></mtr><mtr><mtd><mn>0</mn><mspace></mspace><mtext>if</mtext><mspace></mspace><mi>u</mi><mo>⁄</mo><mo>∼</mo><mi>v</mi><mo>,</mo><mi>u</mi><mo>∈</mo><mi>U</mi><mo>,</mo><mspace></mspace></mtd></mtr></mtable></mrow></mfenced></mrow></math></span></span></span>and set <span><math><mrow><mi>H</mi><mrow><mo>(</mo><mi>U</mi><mo>)</mo></mrow><mo>≔</mo><mrow><mo>{</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>:</mo><mi>v</mi><mo>∈</mo><mi>U</mi><mo>}</mo></mrow></mrow></math></span>. The first purpose of this paper is to study the following question: What families of graphs <span><math><mi>G</mi></math></span> and what conditions on <span><math><mi>U</mi></math></span> do we need so that the VC-dimension of <span><math><mrow><mi>H</mi><mrow><mo>(</mo><mi>U</mi><mo>)</mo></mrow></mrow></math></span> can be determined? We show that if <span><math><mi>G</mi></math></span> is a pseudo-random graph, then under some mild conditions, the VC dimension of <span><math><mrow><mi>H</mi><mrow><mo>(</mo><mi>U</mi><mo>)</mo></mrow></mrow></math></span> can be bounded from below. Specific cases of this theorem recover and improve previous results on VC-dimension of functions defined by the well-studied distance and dot-product graphs over a finite field.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"365 ","pages":"Pages 231-246"},"PeriodicalIF":1.0,"publicationDate":"2025-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150700","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}
引用次数: 0
On the edge-connectivity of the square of a graph
IF 1
3区 数学
Camino Balbuena , Peter Dankelmann
{"title":"On the edge-connectivity of the square of a graph","authors":"Camino Balbuena , Peter Dankelmann","doi":"10.1016/j.dam.2025.01.029","DOIUrl":"10.1016/j.dam.2025.01.029","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a connected graph. The edge-connectivity of <span><math><mi>G</mi></math></span>, denoted by <span><math><mrow><mi>λ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, is the minimum number of edges whose removal renders <span><math><mi>G</mi></math></span> disconnected. Let <span><math><mrow><mi>δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> be the minimum degree of <span><math><mi>G</mi></math></span>. It is well-known that <span><math><mrow><mi>λ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mi>δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, and graphs for which equality holds are said to be maximally edge-connected. The square <span><math><msup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> of <span><math><mi>G</mi></math></span> is the graph with the same vertex set as <span><math><mi>G</mi></math></span>, in which two vertices are adjacent if their distance is not more that 2.</div><div>In this paper we present results on the edge-connectivity of the square of a graph. We show that if the minimum degree of a connected graph <span><math><mi>G</mi></math></span> of order <span><math><mi>n</mi></math></span> is at least <span><math><mrow><mo>⌊</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>⌋</mo></mrow></math></span>, then <span><math><msup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> is maximally edge-connected, and this result is best possible. We also give lower bounds on <span><math><mrow><mi>λ</mi><mrow><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> for the case that <span><math><msup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> is not maximally edge-connected: We prove that <span><math><mrow><mi>λ</mi><mrow><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mo>≥</mo><mi>κ</mi><msup><mrow><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>κ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><mi>κ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> denotes the connectivity of <span><math><mi>G</mi></math></span>, i.e., the minimum number of vertices whose removal renders <span><math><mi>G</mi></math></span> disconnected, and this bound is sharp. We further prove that <span><math><mrow><mi>λ</mi><mrow><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mo>≥</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi>λ</mi><msup><mrow><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi>λ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></m","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"366 ","pages":"Pages 250-256"},"PeriodicalIF":1.0,"publicationDate":"2025-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143161757","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}
引用次数: 0
General sum-connectivity index of unicyclic graphs with given maximum degree
IF 1
3区 数学
Elize Swartz, Tomáš Vetrík
{"title":"General sum-connectivity index of unicyclic graphs with given maximum degree","authors":"Elize Swartz, Tomáš Vetrík","doi":"10.1016/j.dam.2025.01.033","DOIUrl":"10.1016/j.dam.2025.01.033","url":null,"abstract":"<div><div>For <span><math><mrow><mi>a</mi><mo>∈</mo><mi>R</mi></mrow></math></span>, the general sum-connectivity index <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>a</mi></mrow></msub></math></span> of a graph <span><math><mi>G</mi></math></span> is defined as <span><math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>a</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>u</mi><mi>v</mi><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></msub><msup><mrow><mrow><mo>[</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>]</mo></mrow></mrow><mrow><mi>a</mi></mrow></msup></mrow></math></span>, where <span><math><mrow><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is the set of edges of <span><math><mi>G</mi></math></span>, and <span><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></math></span> are the degrees of vertices <span><math><mi>u</mi></math></span> and <span><math><mi>v</mi></math></span>, respectively. Among unicyclic graphs with given number of vertices and maximum degree, we present graphs having the largest and smallest values of <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>a</mi></mrow></msub></math></span>, and we state cases which are still open. We also solve one of the open problems on <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>a</mi></mrow></msub></math></span> for trees if <span><math><mrow><mn>0</mn><mo><</mo><mi>a</mi><mo><</mo><mn>1</mn></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"366 ","pages":"Pages 238-249"},"PeriodicalIF":1.0,"publicationDate":"2025-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143160549","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}
引用次数: 0
Functions that are uniquely maximized by sparse quasi-star graphs, and uniquely minimized by quasi-complete graphs
IF 1
3区 数学
Nicola Apollonio
{"title":"Functions that are uniquely maximized by sparse quasi-star graphs, and uniquely minimized by quasi-complete graphs","authors":"Nicola Apollonio","doi":"10.1016/j.dam.2025.01.032","DOIUrl":"10.1016/j.dam.2025.01.032","url":null,"abstract":"<div><div>We show that for a certain class of convex functions <span><math><mi>f</mi></math></span>, including the exponential functions <span><math><mrow><mi>x</mi><mo>↦</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>λ</mi><mi>x</mi></mrow></msup></mrow></math></span> with <span><math><mrow><mi>λ</mi><mo>></mo><mn>0</mn></mrow></math></span> a real number, and all the powers <span><math><mrow><mi>x</mi><mo>↦</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></math></span>, <span><math><mrow><mi>x</mi><mo>≥</mo><mn>0</mn></mrow></math></span> and <span><math><mrow><mi>β</mi><mo>≥</mo><mn>2</mn></mrow></math></span> a real number, with a unique small exception, if <span><math><mrow><mo>(</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></math></span> ranges over the degree sequences of graphs with <span><math><mi>n</mi></math></span> vertices and <span><math><mi>m</mi></math></span> edges and <span><math><mrow><mi>m</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></math></span>, then the maximum of <span><math><mrow><msub><mrow><mo>∑</mo></mrow><mrow><mi>i</mi></mrow></msub><mi>f</mi><mrow><mo>(</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> is uniquely attained by the degree sequence of a quasi-star graph, namely, a graph consisting of a star plus possibly additional isolated vertices. This result significantly extends a similar result in Ismailescu and Stefanica (2002). Dually, we show that for a certain class of concave functions <span><math><mi>g</mi></math></span>, including the negative exponential functions <span><math><mrow><mi>x</mi><mo>↦</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mi>λ</mi><mi>x</mi></mrow></msup></mrow></math></span> with <span><math><mrow><mi>λ</mi><mo>></mo><mo>ln</mo><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span> a real number, all the powers <span><math><mrow><mi>x</mi><mo>↦</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></math></span>, <span><math><mrow><mi>x</mi><mo>≥</mo><mn>0</mn></mrow></math></span> and <span><math><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo>≤</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span> a real number, and the function <span><math><mrow><mi>x</mi><mo>↦</mo><mfrac><mrow><mi>x</mi></mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></math></span> for <span><math><mrow><mi>x</mi><mo>≥</mo><mn>0</mn></mrow></math></span>, if <span><math><mrow><mo>(</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></math></span> ranges over the degree sequences of graphs with <span><math><mi>n</mi></math></span> vertices and <span><math><mi>m</mi></math></span> edges, then the minim","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"366 ","pages":"Pages 226-237"},"PeriodicalIF":1.0,"publicationDate":"2025-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143161752","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}
引用次数: 0
Coloring of (P6,dart, K4)-free graphs
IF 1
3区 数学
Xia Hong , Baogang Xu
{"title":"Coloring of (P6,dart, K4)-free graphs","authors":"Xia Hong , Baogang Xu","doi":"10.1016/j.dam.2025.01.034","DOIUrl":"10.1016/j.dam.2025.01.034","url":null,"abstract":"<div><div>A hereditary class <span><math><mi>G</mi></math></span> of graphs is <span><math><mi>χ</mi></math></span>-bounded if there is a <span><math><mi>χ</mi></math></span>-binding function, say <span><math><mi>f</mi></math></span>, such that <span><math><mrow><mi>χ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mi>f</mi><mrow><mo>(</mo><mi>ω</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span>, for every <span><math><mrow><mi>G</mi><mo>∈</mo><mi>G</mi></mrow></math></span>, where <span><math><mrow><mi>χ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mspace></mspace><mrow><mo>(</mo><mi>ω</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> denotes the chromatic (clique) number of <span><math><mi>G</mi></math></span>. A <em>diamond</em> is a graph obtained from <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> by removing an edge, and a <em>dart</em> is a graph obtained from a <em>diamond</em> by adding a pendent edge to a vertex of degree 3 of the <em>diamond</em>. Gravier, Hoáng and Maffray showed that every (<span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>6</mn></mrow></msub></math></span>, <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>)-free graph is 16-colorable (Gravier et al., 2003). Brause and Gei<span><math><mi>β</mi></math></span>er showed that (<span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span>, <em>dart</em>)-free graphs <span><math><mi>G</mi></math></span> have a binding function <span><math><mrow><mi>Θ</mi><mrow><mo>(</mo><mfrac><mrow><msup><mrow><mi>ω</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow><mrow><mo>log</mo><mi>ω</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></mfrac><mo>)</mo></mrow></mrow></math></span> (Brause and Gei<span><math><mi>β</mi></math></span>er, 2021). In this paper, we show that every (<span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>6</mn></mrow></msub></math></span>, <em>dart</em>, <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>)-free graph is 7-colorable.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"365 ","pages":"Pages 223-230"},"PeriodicalIF":1.0,"publicationDate":"2025-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150699","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}
引用次数: 0
Cubic and quartic net-regular strongly regular signed graphs
IF 1
3区 数学
Milica Anđelić , Tamara Koledin , Zoran Stanić
{"title":"Cubic and quartic net-regular strongly regular signed graphs","authors":"Milica Anđelić , Tamara Koledin , Zoran Stanić","doi":"10.1016/j.dam.2025.01.026","DOIUrl":"10.1016/j.dam.2025.01.026","url":null,"abstract":"<div><div>In this paper, we determine all connected net-regular strongly regular signed graphs with vertex degree at most four.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"366 ","pages":"Pages 216-225"},"PeriodicalIF":1.0,"publicationDate":"2025-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143160551","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}
引用次数: 0
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