{"title":"Rainbow Hamiltonicity and the spectral radius","authors":"Yuke Zhang , 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}
Reza Akhtar , Jacob Charboneau , Stephen M. Gagola III
{"title":"Strong complete mappings for 2-groups","authors":"Reza Akhtar , Jacob Charboneau , 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}
Marcin Anholcer , Bartłomiej Bosek , Jarosław Grytczuk , Grzegorz Gutowski , Jakub Przybyło , Mariusz Zając
{"title":"Mrs. Correct and majority colorings","authors":"Marcin Anholcer , Bartłomiej Bosek , Jarosław Grytczuk , Grzegorz Gutowski , Jakub Przybyło , 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 , Ao Fan , Jinlong Shu","doi":"10.1016/j.disc.2025.114593","DOIUrl":"10.1016/j.disc.2025.114593","url":null,"abstract":"<div><div>The <em>toughness</em> <span><math><mi>τ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mspace></mspace><mrow><mi>min</mi></mrow><mo>{</mo><mfrac><mrow><mo>|</mo><mi>S</mi><mo>|</mo></mrow><mrow><mi>c</mi><mo>(</mo><mi>G</mi><mo>−</mo><mi>S</mi><mo>)</mo></mrow></mfrac><mo>:</mo><mi>S</mi><mspace></mspace><mtext>is a cut set of vertices in</mtext><mspace></mspace><mi>G</mi><mo>}</mo></math></span> for <span><math><mi>G</mi><mo>≇</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, which was initially proposed by Chvátal in 1973. A graph <em>G</em> is called <em>t-tough</em> if <span><math><mi>τ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mi>t</mi></math></span>. 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?</div><div>A <em>spanning k-tree</em> of a connected graph <em>G</em> is a spanning tree with the degree of every vertex at most <em>k</em>, which is considered as a connected <span><math><mo>[</mo><mn>1</mn><mo>,</mo><mi>k</mi><mo>]</mo></math></span>-factor. We in this paper provide a tight sufficient condition based on the spectral radius for a connected <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>k</mi><mo>−</mo><mi>η</mi></mrow></mfrac></math></span>-tough graph to contain a spanning <em>k</em>-tree, where <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span> is an integer and <span><math><mi>η</mi><mo>=</mo><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>.</div><div>Let <span><math><mi>b</mi><mo>≥</mo><mn>1</mn></math></span> be an integer. An <em>odd</em> <span><math><mo>[</mo><mn>1</mn><mo>,</mo><mi>b</mi><mo>]</mo></math></span><em>-factor</em> of a graph <em>G</em> is a spanning subgraph <em>F</em> such that for each <span><math><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>F</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo></math></span> is odd and <span><math><mn>1</mn><mo>≤</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>F</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo><mo>≤</mo><mi>b</mi></math></span>. We propose a tight sufficient condition in terms of the spectral radius for a connected <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>b</mi><mo>+</mo><mn>1</mn></mrow></mfrac></math></span>-tough graph to contain an odd <span><math><mo>[</mo><mn>1</mn><mo>,</mo><mi>b</mi><mo>]</mo></math></span>-factor. If <span><math><mi>b</mi><mo>=</mo><mn>1</mn></math></span>, an odd <span><math><mo>[</mo><mn>1</mn><mo>,</mo><mi>b</mi><mo>]</mo></math></span>-factor is called a 1-factor (perfect matching). We also present a tight sufficient condition in terms of the spectral radius for a connected <span><math><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, 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}
{"title":"An extension of spectral Mantel's theorem on wheels","authors":"Rui Li , Bo Liu , Mingqing Zhai","doi":"10.1016/j.disc.2025.114573","DOIUrl":"10.1016/j.disc.2025.114573","url":null,"abstract":"<div><div>A graph is considered wheel-free if the neighborhood of any vertex is acyclic. The extremal problems associated with wheel-free graphs have a long-standing history of research. In 1983, Gallai and Zelinka independently posed the question of determining the maximum number of triangles in an <em>n</em>-vertex wheel-free graph. Moving forward to 2021, Zhao, Huang and Lin investigated the maximum spectral radius of graphs within the same family of wheel-free graphs.</div><div>In this paper, we focus on graphs of fixed size that do not contain isolated vertices. In 1970, Nosal established that every graph <em>G</em> with <em>m</em> edges and a spectral radius <span><math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>></mo><msqrt><mrow><mi>m</mi></mrow></msqrt></math></span> contains at least one triangle. This result is known as the spectral Mantel's theorem. Nikiforov further refined this theorem by showing that any graph <em>G</em> with <em>m</em> edges and a spectral radius <span><math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><msqrt><mrow><mi>m</mi></mrow></msqrt></math></span> contains a triangle, except when <em>G</em> is a complete bipartite graph.</div><div>In this work, we present an extension of spectral Mantel's theorem, which asserts that every graph <em>G</em> with <span><math><mi>m</mi><mo>≥</mo><mn>25</mn></math></span> edges and a spectral radius <span><math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>(</mo><mn>1</mn><mo>+</mo><msqrt><mrow><mn>4</mn><mi>m</mi><mo>−</mo><mn>5</mn></mrow></msqrt><mo>)</mo></math></span> contains a wheel, unless <em>G</em> is a book (possibly missing an edge).</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 11","pages":"Article 114573"},"PeriodicalIF":0.7,"publicationDate":"2025-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144105957","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":"Oriented Hamiltonian paths in tournaments with an arc removed","authors":"Ayman El Zein","doi":"10.1016/j.disc.2025.114578","DOIUrl":"10.1016/j.disc.2025.114578","url":null,"abstract":"<div><div>Havet and Thomassé settled Rosenfeld's conjecture, which says that any tournament of order <span><math><mi>n</mi><mo>≥</mo><mn>8</mn></math></span> contains any oriented Hamiltonian path. In this paper, we show that in any tournament <em>T</em> of order <span><math><mi>n</mi><mo>≥</mo><mn>8</mn></math></span>, there exists an arc <em>e</em> such that <span><math><mi>T</mi><mo>−</mo><mi>e</mi></math></span> contains any oriented Hamiltonian path.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 11","pages":"Article 114578"},"PeriodicalIF":0.7,"publicationDate":"2025-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144105862","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 real symmetric matrices with a given rank and a P-set with maximum size","authors":"Zhibin Du , Carlos M. da Fonseca","doi":"10.1016/j.disc.2025.114572","DOIUrl":"10.1016/j.disc.2025.114572","url":null,"abstract":"<div><div>Given a real symmetric matrix <em>A</em> of rank <em>r</em>, the maximum size of a P-set of <em>A</em> does not exceed <span><math><mo>⌊</mo><mi>r</mi><mo>/</mo><mn>2</mn><mo>⌋</mo></math></span>. In this work, we fully characterize the maximal graphs for which there is a real symmetric matrix attaining this bound.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 11","pages":"Article 114572"},"PeriodicalIF":0.7,"publicationDate":"2025-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144114992","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":"Difference graphs of étale algebras over finite fields","authors":"Neha Prabhu","doi":"10.1016/j.disc.2025.114569","DOIUrl":"10.1016/j.disc.2025.114569","url":null,"abstract":"<div><div>Properties of difference graphs of finite fields <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> were studied by Winnie Li in 1992. This article extends her work and investigates the spectrum, connectivity, girth of difference graphs of étale algebras over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. The study covers fields of odd as well as even characteristic, and identifies the bipartite graphs in this family. Explicit examples of non-Ramanujan graphs are obtained.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 11","pages":"Article 114569"},"PeriodicalIF":0.7,"publicationDate":"2025-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144099332","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":"Digraphs in which every t vertices have exactly λ common out-neighbors","authors":"Myungho Choi , Hojin Chu , Suh-Ryung Kim","doi":"10.1016/j.disc.2025.114580","DOIUrl":"10.1016/j.disc.2025.114580","url":null,"abstract":"<div><div>We say that a digraph is a <span><math><mo>(</mo><mi>t</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math></span>-liking digraph if every <em>t</em> vertices have exactly <em>λ</em> common out-neighbors. In 1975, Plesník (1975) <span><span>[14]</span></span> proved that any <span><math><mo>(</mo><mi>t</mi><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-liking digraph is the complete digraph on <span><math><mi>t</mi><mo>+</mo><mn>1</mn></math></span> vertices for each <span><math><mi>t</mi><mo>≥</mo><mn>3</mn></math></span>. Choi et al. (2025) <span><span>[5]</span></span> showed that a <span><math><mo>(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-liking digraph is a fancy wheel digraph or a <em>k</em>-diregular digraph for some positive integer <em>k</em>. In this paper, we extend these results by completely characterizing the <span><math><mo>(</mo><mi>t</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math></span>-liking digraphs with <span><math><mi>t</mi><mo>≥</mo><mi>λ</mi><mo>+</mo><mn>2</mn></math></span> and giving some equivalent conditions for a <span><math><mo>(</mo><mi>t</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math></span>-liking digraph being a complete digraph on <span><math><mi>t</mi><mo>+</mo><mi>λ</mi></math></span> vertices.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 10","pages":"Article 114580"},"PeriodicalIF":0.7,"publicationDate":"2025-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144099603","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}
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
