Discrete Mathematics最新文献

{"title":"Forest cuts in sparse graphs","authors":"Vsevolod Chernyshev ,&nbsp;Johannes Rauch ,&nbsp;Dieter Rautenbach","doi":"10.1016/j.disc.2025.114594","DOIUrl":"10.1016/j.disc.2025.114594","url":null,"abstract":"<div><div>We consider the conjecture that every graph <em>G</em> of order <em>n</em> with less than <span><math><mn>3</mn><mi>n</mi><mo>−</mo><mn>6</mn></math></span> edges has a vertex cut that induces a forest. Maximal planar graphs do not have such vertex cuts and show that the density condition would be best possible. We verify the conjecture for planar graphs and show that every graph <em>G</em> of order <em>n</em> with less than <span><math><mfrac><mrow><mn>11</mn></mrow><mrow><mn>5</mn></mrow></mfrac><mi>n</mi><mo>−</mo><mfrac><mrow><mn>18</mn></mrow><mrow><mn>5</mn></mrow></mfrac></math></span> edges has a vertex cut that induces a forest.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 11","pages":"Article 114594"},"PeriodicalIF":0.7,"publicationDate":"2025-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144137780","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 rainbow numbers of paths in maximal bipartite planar graphs","authors":"Lei Ren,&nbsp;Yongxin Lan,&nbsp;Changqing Xu","doi":"10.1016/j.disc.2025.114596","DOIUrl":"10.1016/j.disc.2025.114596","url":null,"abstract":"<div><div>Given two graphs <em>G</em> and <em>T</em>, the rainbow number of <em>T</em> in <em>G</em>, denoted by <span><math><mi>r</mi><mi>b</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>T</mi><mo>)</mo></math></span>, is the minimum positive integer <em>t</em> such that, if <em>G</em> contains a copy of <em>T</em>, then every <em>t</em>-edge-coloring of <em>G</em> contains a rainbow copy of <em>T</em>. Given a family of graphs <span><math><mi>G</mi></math></span> and a graph <em>T</em>, if every graph in <span><math><mi>G</mi></math></span> contains a copy of <em>T</em>, then the rainbow number of <em>T</em> in <span><math><mi>G</mi></math></span>, denoted by <span><math><mi>r</mi><mi>b</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>T</mi><mo>)</mo></math></span>, is defined as <span><math><mi>max</mi><mo>⁡</mo><mo>{</mo><mi>r</mi><mi>b</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>T</mi><mo>)</mo><mo>|</mo><mi>G</mi><mo>∈</mo><mi>G</mi><mo>}</mo></math></span>. Given a graph <em>T</em>, let <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>T</mi><mo>)</mo></math></span> denote the family of all maximal bipartite planar graphs on <em>n</em> vertices that contain a copy of <em>T</em>. In this paper, we study the rainbow numbers of paths in maximal bipartite planar graphs, we get the exact value of <span><math><mi>r</mi><mi>b</mi><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></math></span> for <span><math><mi>n</mi><mo>≥</mo><mi>ℓ</mi></math></span> and <span><math><mi>ℓ</mi><mo>≠</mo><mn>8</mn></math></span>, and the lower bound of <span><math><mi>r</mi><mi>b</mi><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>8</mn></mrow></msub><mo>)</mo><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>8</mn></mrow></msub><mo>)</mo></math></span> for all <span><math><mi>n</mi><mo>≥</mo><mn>8</mn></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 11","pages":"Article 114596"},"PeriodicalIF":0.7,"publicationDate":"2025-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144134929","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":"Pattern avoidance in revised ascent sequences","authors":"Robin D.P. Zhou","doi":"10.1016/j.disc.2025.114608","DOIUrl":"10.1016/j.disc.2025.114608","url":null,"abstract":"<div><div>Inspired by the definition of modified ascent sequences, we introduce a new class of integer sequences called revised ascent sequences. These sequences are defined as Cayley permutations where each entry is a leftmost occurrence if and only if it serves as an ascent bottom. We construct a bijection between ascent sequences and revised ascent sequences by adapting the classic hat map, which transforms ascent sequences into modified ascent sequences. Additionally, we investigate revised ascent sequences that avoid a single pattern, leading to a wealth of enumerative results. Our main techniques include the use of bijections, generating trees, generating functions, and the kernel method.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 11","pages":"Article 114608"},"PeriodicalIF":0.7,"publicationDate":"2025-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144134931","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":"Enumerating several statistics of r-colored Dyck paths with no dd-steps having the same colors","authors":"Yidong Sun,&nbsp;Jinyi Wang,&nbsp;Xinyu Wang","doi":"10.1016/j.disc.2025.114597","DOIUrl":"10.1016/j.disc.2025.114597","url":null,"abstract":"<div><div>An <em>r</em>-colored Dyck path is a Dyck path with all <strong>d</strong>-steps having one of <em>r</em> colors in <span><math><mo>[</mo><mi>r</mi><mo>]</mo><mo>=</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>r</mi><mo>}</mo></math></span>. In this paper, we consider several statistics on the set <span><math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mo>,</mo><mn>0</mn></mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></msubsup></math></span> of <em>r</em>-colored Dyck paths of length 2<em>n</em> with no two consecutive <strong>d</strong>-steps having the same colors. Precisely, the paper studies the statistics “number of points” at level <em>ℓ</em>, “number of <strong>u</strong>-steps” at level <span><math><mi>ℓ</mi><mo>+</mo><mn>1</mn></math></span>, “number of peaks” at level <span><math><mi>ℓ</mi><mo>+</mo><mn>1</mn></math></span> and “number of <strong>udu</strong>-steps” on the set <span><math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mo>,</mo><mn>0</mn></mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></msubsup></math></span>. The counting formulas of the first three statistics are established by Riordan arrays related to <span><math><mi>S</mi><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>;</mo><mi>x</mi><mo>)</mo></math></span>, the weighted generating function of <span><math><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></math></span>-Schröder paths. By a useful and surprising relations satisfied by <span><math><mi>S</mi><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>;</mo><mi>x</mi><mo>)</mo></math></span>, several identities related to these counting formulas are also described.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 11","pages":"Article 114597"},"PeriodicalIF":0.7,"publicationDate":"2025-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144134930","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":"Distance-based (and path-based) covering problems for graphs of given cyclomatic number","authors":"Dibyayan Chakraborty ,&nbsp;Florent Foucaud ,&nbsp;Anni Hakanen","doi":"10.1016/j.disc.2025.114595","DOIUrl":"10.1016/j.disc.2025.114595","url":null,"abstract":"<div><div>We study a large family of graph covering problems, whose definitions rely on distances, for graphs of bounded cyclomatic number (that is, the minimum number of edges that need to be removed from the graph to destroy all cycles). These problems include (but are not restricted to) three families of problems: (i) variants of metric dimension, where one wants to choose a small set <em>S</em> of vertices of the graph such that every vertex is uniquely determined by its ordered vector of distances to the vertices of <em>S</em>; (ii) variants of geodetic sets, where one wants to select a small set <em>S</em> of vertices such that any vertex lies on some shortest path between two vertices of <em>S</em>; (iii) variants of path covers, where one wants to select a small set of paths such that every vertex or edge belongs to one of the paths. We generalize and/or improve previous results in the area which show that the optimal values for these problems can be upper-bounded by a linear function of the cyclomatic number and the degree 1-vertices of the graph. To this end, we develop and enhance a technique recently introduced in (Lu et al., 2022 <span><span>[53]</span></span>) and give near-optimal bounds in several cases. This solves (in some cases fully, in some cases partially) some conjectures and open questions from the literature. The method, based on breadth-first search, is of algorithmic nature and thus, all the constructions can be computed in linear time. Our results also imply an algorithmic consequence for the computation of the <em>optimal</em> solutions: for some of the problems, they can be computed in polynomial time for graphs of bounded cyclomatic number.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 11","pages":"Article 114595"},"PeriodicalIF":0.7,"publicationDate":"2025-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144116592","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":"Rainbow Hamiltonicity and the spectral radius","authors":"Yuke Zhang ,&nbsp;Edwin R. van Dam","doi":"10.1016/j.disc.2025.114600","DOIUrl":"10.1016/j.disc.2025.114600","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></math></span> be a family of graphs of order <em>n</em> with the same vertex set. A rainbow Hamiltonian cycle in <span><math><mi>G</mi></math></span> is a cycle that visits each vertex precisely once such that any two edges belong to different graphs of <span><math><mi>G</mi></math></span>. We show that if each <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> has more than <span><math><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mn>1</mn></math></span> edges, then <span><math><mi>G</mi></math></span> admits a rainbow Hamiltonian cycle and pose the problem of characterizing rainbow Hamiltonicity under the condition that all <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> have at least <span><math><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mn>1</mn></math></span> edges. Towards a solution of that problem, we give a sufficient condition for the existence of a rainbow Hamiltonian cycle in terms of the spectral radii of the graphs in <span><math><mi>G</mi></math></span> and completely characterize the corresponding extremal graphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 11","pages":"Article 114600"},"PeriodicalIF":0.7,"publicationDate":"2025-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144116593","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":"Strong complete mappings for 2-groups","authors":"Reza Akhtar ,&nbsp;Jacob Charboneau ,&nbsp;Stephen M. Gagola III","doi":"10.1016/j.disc.2025.114568","DOIUrl":"10.1016/j.disc.2025.114568","url":null,"abstract":"<div><div>A strong complete mapping for a group <em>G</em> is a bijection <span><math><mi>φ</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>G</mi></math></span> such that the maps <span><math><mi>x</mi><mo>↦</mo><mi>x</mi><mi>φ</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> and <span><math><mi>x</mi><mo>↦</mo><msup><mrow><mi>x</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>φ</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> are also bijections. Groups admitting a strong complete mapping are important to the study of orthogonality problems for Latin squares and group sequencings, among other applications. In previous work we showed that a finite 3-group that contains no cyclic subgroup of index 3 is strongly admissible. In this article, we employ a substantially different strategy to show that a finite 2-group that contains no cyclic subgroup of index 4 is strongly admissible.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 10","pages":"Article 114568"},"PeriodicalIF":0.7,"publicationDate":"2025-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144116891","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":"Mrs. Correct and majority colorings","authors":"Marcin Anholcer ,&nbsp;Bartłomiej Bosek ,&nbsp;Jarosław Grytczuk ,&nbsp;Grzegorz Gutowski ,&nbsp;Jakub Przybyło ,&nbsp;Mariusz Zając","doi":"10.1016/j.disc.2025.114577","DOIUrl":"10.1016/j.disc.2025.114577","url":null,"abstract":"<div><div>A <em>majority coloring</em> of a directed graph is a vertex coloring in which each vertex has the same color as at most half of its out-neighbors. In this note we simplify some proof techniques and generalize previously known results on various generalizations of majority coloring. In particular, our unified and simplified approach works for <em>paintability</em> – an on-line analog of the list coloring.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 11","pages":"Article 114577"},"PeriodicalIF":0.7,"publicationDate":"2025-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144114993","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":"Spectral extremal problems on factors in tough graphs, and beyond","authors":"Ruifang Liu ,&nbsp;Ao Fan ,&nbsp;Jinlong Shu","doi":"10.1016/j.disc.2025.114593","DOIUrl":"10.1016/j.disc.2025.114593","url":null,"abstract":"&lt;div&gt;&lt;div&gt;The &lt;em&gt;toughness&lt;/em&gt; &lt;span&gt;&lt;math&gt;&lt;mi&gt;τ&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mrow&gt;&lt;mi&gt;min&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mtext&gt;is a cut set of vertices in&lt;/mtext&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;≇&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, which was initially proposed by Chvátal in 1973. A graph &lt;em&gt;G&lt;/em&gt; is called &lt;em&gt;t-tough&lt;/em&gt; if &lt;span&gt;&lt;math&gt;&lt;mi&gt;τ&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. Fan, Lin and Lu [European J. Combin. 110 (2023) 103701] presented a tight sufficient condition in terms of the spectral radius for a connected 1-tough graph to contain a connected 2-factor (Hamilton cycle). A natural and interesting problem arises: What is a tight spectral condition to guarantee the existence of factors among tough graphs?&lt;/div&gt;&lt;div&gt;A &lt;em&gt;spanning k-tree&lt;/em&gt; of a connected graph &lt;em&gt;G&lt;/em&gt; is a spanning tree with the degree of every vertex at most &lt;em&gt;k&lt;/em&gt;, which is considered as a connected &lt;span&gt;&lt;math&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-factor. We in this paper provide a tight sufficient condition based on the spectral radius for a connected &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;η&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt;-tough graph to contain a spanning &lt;em&gt;k&lt;/em&gt;-tree, where &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; is an integer and &lt;span&gt;&lt;math&gt;&lt;mi&gt;η&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;.&lt;/div&gt;&lt;div&gt;Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; be an integer. An &lt;em&gt;odd&lt;/em&gt; &lt;span&gt;&lt;math&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;em&gt;-factor&lt;/em&gt; of a graph &lt;em&gt;G&lt;/em&gt; is a spanning subgraph &lt;em&gt;F&lt;/em&gt; such that for each &lt;span&gt;&lt;math&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is odd and &lt;span&gt;&lt;math&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. We propose a tight sufficient condition in terms of the spectral radius for a connected &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt;-tough graph to contain an odd &lt;span&gt;&lt;math&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-factor. If &lt;span&gt;&lt;math&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, an odd &lt;span&gt;&lt;math&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-factor is called a 1-factor (perfect matching). We also present a tight sufficient condition in terms of the spectral radius for a connected &lt;span&gt;&lt;math&gt;&lt;mfr","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 11","pages":"Article 114593"},"PeriodicalIF":0.7,"publicationDate":"2025-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144105868","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":"Bounds for the smallest positive eigenvalue of unicyclic graphs with diameter at most 4","authors":"Sasmita Barik,&nbsp;Piyush Verma","doi":"10.1016/j.disc.2025.114574","DOIUrl":"10.1016/j.disc.2025.114574","url":null,"abstract":"<div><div>Let <em>G</em> be a simple graph on <em>n</em> vertices and <span><math><mi>τ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> denote the smallest positive eigenvalue of its adjacency matrix <span><math><mi>A</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. In [S. Rani and S. Barik, Upper bounds on the smallest positive eigenvalues of trees, Ann. Comb. 27(1) (2023) 19–29], the authors characterized the trees with small diameters having the maximum and minimum <em>τ</em>, respectively. In this article, we extend their work to the unicyclic graphs. We provide bounds for the smallest positive eigenvalue and obtain the graphs with the maximum and minimum <em>τ</em> among all the unicyclic graphs on <em>n</em> vertices having diameters 2 and 3, respectively. Furthermore, we characterize the graphs with the maximum <em>τ</em> among all the unicyclic graphs on <em>n</em> vertices having diameter 4. Finally, we characterize all the unicyclic graphs on <em>n</em> vertices with diameter at most 4 whose smallest positive eigenvalue is equal to <span><math><mfrac><mrow><msqrt><mrow><mn>5</mn></mrow></msqrt><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>, the reciprocal of the golden ratio.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 11","pages":"Article 114574"},"PeriodicalIF":0.7,"publicationDate":"2025-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144105869","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}