{"title":"On the structure of perfectly divisible graphs","authors":"Chính T. Hoàng","doi":"10.1016/j.disc.2025.114809","DOIUrl":"10.1016/j.disc.2025.114809","url":null,"abstract":"<div><div>A graph <em>G</em> is perfectly divisible if every induced subgraph <em>H</em> of <em>G</em> contains a set <em>X</em> of vertices such that <em>X</em> meets all largest cliques of <em>H</em>, and <em>X</em> induces a perfect graph. The chromatic number of a perfectly divisible graph <em>G</em> is bounded by <span><math><msup><mrow><mi>ω</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> where <em>ω</em> denotes the number of vertices in a largest clique of <em>G</em>. A graph <em>G</em> is minimally non-perfectly divisible if <em>G</em> is not perfectly divisible but each of its proper induced subgraph is. A set <em>C</em> of vertices of <em>G</em> is a clique cutset if <em>C</em> induces a clique in <em>G</em>, and <span><math><mi>G</mi><mo>−</mo><mi>C</mi></math></span> is disconnected. We prove that a <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span>-free minimally non-perfectly divisible graph cannot contain a clique cutset. This result allows us to re-establish several theorems on the perfect divisibility of some classes of <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span>-free graphs. We will show that recognizing perfectly divisible graphs is NP-hard.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114809"},"PeriodicalIF":0.7,"publicationDate":"2025-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145157628","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":"Generalized Cayley graphs and perfect code","authors":"Fateme Sadat Seiedali , Zeinab Akhlaghi , Behrooz Khosravi","doi":"10.1016/j.disc.2025.114805","DOIUrl":"10.1016/j.disc.2025.114805","url":null,"abstract":"<div><div>A subset <em>C</em> of the vertex set of a graph Γ is said to be <span><math><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></math></span>-regular if <em>C</em> induces an <em>a</em>-regular subgraph and every vertex outside <em>C</em> is adjacent to exactly <em>b</em> vertices in <em>C</em>. A <span><math><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-regular set is called a perfect code. Let <em>G</em> be a group and <span><math><mi>α</mi><mo>∈</mo><mrow><mi>Aut</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></math></span> such that <span><math><msup><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mrow><mi>id</mi></mrow></math></span>. Let <span><math><mi>S</mi><mo>⊆</mo><mi>G</mi></math></span>, with <span><math><mi>α</mi><mo>(</mo><mi>S</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>S</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> and <span><math><mi>S</mi><mo>∩</mo><mo>{</mo><mi>α</mi><mo>(</mo><msup><mrow><mi>g</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo><mi>g</mi><mspace></mspace><mo>:</mo><mspace></mspace><mi>g</mi><mo>∈</mo><mi>G</mi><mo>}</mo><mo>=</mo><mo>∅</mo></math></span>. The generalized Cayley graph of <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>α</mi><mo>)</mo></math></span> with respect to <em>S</em> is a graph with vertex set <em>G</em> and two distinct elements <span><math><mi>g</mi><mo>,</mo><mi>h</mi><mo>∈</mo><mi>G</mi></math></span> are adjacent if and only if <span><math><mi>α</mi><mo>(</mo><msup><mrow><mi>g</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo><mi>h</mi><mo>∈</mo><mi>S</mi></math></span>. If <span><math><mi>α</mi><mo>=</mo><mrow><mi>id</mi></mrow></math></span>, then the described graph is called a Cayley graph of <em>G</em> with respect to <em>S</em>. By an <span><math><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></math></span>-regular set (resp. a perfect code) of <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>α</mi><mo>)</mo></math></span> we mean an <span><math><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></math></span>-regular set (resp. a perfect code) in a generalized Cayley graph of <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>α</mi><mo>)</mo></math></span> with respect to some subset <em>S</em>. Let <em>G</em> be a group, <span><math><mi>α</mi><mo>∈</mo><mrow><mi>Aut</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></math></span>, <span><math><msup><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mrow><mi>id</mi></mrow></math></span> and <em>H</em> be a subgroup of <em>G</em>. In this paper, we give a necessary and sufficient condition for <em>H</em> to be a perfect code of <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>α</mi><mo>)</mo></math></span>. This result is a generalization of <span><span>[9, Theorem 3.1]</span></span> that gives a condition for a subgroup to be a perfect code in a Cayley graph of <em>G</em>. As another result, when <em>G</em> is an abelian group, we determine all pairs <span","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114805"},"PeriodicalIF":0.7,"publicationDate":"2025-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145117681","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}
Stijn Cambie , Ervin Győri , Nika Salia , Casey Tompkins , James Tuite
{"title":"The maximum Wiener index of a uniform hypergraph","authors":"Stijn Cambie , Ervin Győri , Nika Salia , Casey Tompkins , James Tuite","doi":"10.1016/j.disc.2025.114797","DOIUrl":"10.1016/j.disc.2025.114797","url":null,"abstract":"<div><div>The Wiener index of a (hyper)graph is calculated by summing up the distances between all pairs of vertices. We determine the maximum possible Wiener index of a connected <em>n</em>-vertex <em>k</em>-uniform hypergraph and characterize all hypergraphs attaining the maximum Wiener index for every <em>n</em> and <em>k</em>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114797"},"PeriodicalIF":0.7,"publicationDate":"2025-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145119467","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":"Collatz high cycles do not exist","authors":"Kevin Knight","doi":"10.1016/j.disc.2025.114812","DOIUrl":"10.1016/j.disc.2025.114812","url":null,"abstract":"<div><div>The Collatz function takes odd <em>n</em> to <span><math><mo>(</mo><mn>3</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn></math></span> and even <em>n</em> to <span><math><mi>n</mi><mo>/</mo><mn>2</mn></math></span>. Under the iterated Collatz function, every positive integer is conjectured to end up in the trivial cycle 1-2-1. Two types of rational Collatz cycles are of special interest. Consider the set <span><math><mi>S</mi><mo>(</mo><mi>k</mi><mo>,</mo><mi>x</mi><mo>)</mo></math></span> consisting of the smallest members of <em>k</em>-length cycles with <em>x</em> odd terms. The <em>circuit</em> contains the smallest member of <span><math><mi>S</mi><mo>(</mo><mi>k</mi><mo>,</mo><mi>x</mi><mo>)</mo></math></span>, while the <em>high cycle</em> contains the largest. It is known that no circuits of positive integers exist (except 1-2-1); this paper shows that there are likewise no high cycles of positive integers.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114812"},"PeriodicalIF":0.7,"publicationDate":"2025-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145119469","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":"Hamiltonian claw-free graphs with path-type local degree conditions","authors":"Xia Liu, Miao Wang, Shuo Zhang","doi":"10.1016/j.disc.2025.114807","DOIUrl":"10.1016/j.disc.2025.114807","url":null,"abstract":"<div><div>Let <em>k</em> be an integer and <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> be a path on <em>k</em> vertices. For a graph <em>H</em> with an induced <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, define <span><math><msub><mrow><mi>δ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msub><mo>(</mo><mi>H</mi><mo>)</mo><mo>=</mo><mi>min</mi><mo></mo><mo>{</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo></math></span>: <em>v</em> is an end-vertex of an induced <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> of <em>H</em>}. In this paper, we prove that for a 3-connected non-Hamiltonian claw-free graph <em>H</em>, <span><math><msub><mrow><mi>δ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msub><mo>(</mo><mi>H</mi><mo>)</mo><mo>=</mo><mi>δ</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> for any <span><math><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mn>9</mn></math></span>. As by-products, we obtained two results extend the results in Chen et al. (2017) <span><span>[10]</span></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114807"},"PeriodicalIF":0.7,"publicationDate":"2025-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145119472","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":"Toughness, Hamiltonicity and eigenvalues of graphs","authors":"Hongzhang Chen , Jianxi Li , Shou-Jun Xu","doi":"10.1016/j.disc.2025.114806","DOIUrl":"10.1016/j.disc.2025.114806","url":null,"abstract":"<div><div>For a real number <span><math><mi>t</mi><mo>≥</mo><mn>0</mn></math></span>, we say a graph <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span> is <em>t</em>-tough if <span><math><mo>|</mo><mi>S</mi><mo>|</mo><mo>≥</mo><mi>t</mi><mo>⋅</mo><mi>c</mi><mo>(</mo><mi>G</mi><mo>−</mo><mi>S</mi><mo>)</mo></math></span> for all <span><math><mi>S</mi><mo>⊆</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> with <span><math><mi>c</mi><mo>(</mo><mi>G</mi><mo>−</mo><mi>S</mi><mo>)</mo><mo>≥</mo><mn>2</mn></math></span>, where <span><math><mi>c</mi><mo>(</mo><mi>G</mi><mo>−</mo><mi>S</mi><mo>)</mo></math></span> is the number of components of <span><math><mi>G</mi><mo>−</mo><mi>S</mi></math></span>. The toughness <span><math><mi>τ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of <em>G</em> is the maximum <em>t</em> for which <em>G</em> is <em>t</em>-tough. Firstly, we provide a lower bound for <span><math><mi>τ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> in terms of its normalized Laplacian eigenvalues, improving or generalizing known lower bounds established by Huang, Das and Zhu (2022), Gu (2021) and Zhang (2023). We also derive upper bounds for certain eigenvalues in a regular graph to ensure that the graph is <em>t</em>-tough, where <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>t</mi></mrow></mfrac></math></span> is an integer, which extends the related result of Cioabă and Wong (2014). Additionally, we establish a sufficient condition involving the number of <em>r</em>-cliques to ensure the existence of a Hamiltonian cycle in a <em>t</em>-tough graph, where <em>r</em> is an integer with <span><math><mi>r</mi><mo>≥</mo><mn>2</mn></math></span>, which improves upon the sufficient condition based on the number of edges proposed by Cai, Yu, Xu and Yu (2022). Finally, we provide a spectral condition to guarantee the existence of a Hamiltonian cycle in <em>t</em>-tough graphs, thereby addressing the problem posed by Fan, Lin and Lu (2023) for integers <span><math><mi>t</mi><mo>≥</mo><mn>1</mn></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114806"},"PeriodicalIF":0.7,"publicationDate":"2025-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145119471","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":"Partite saturation number of cycles","authors":"Yiduo Xu , Zhen He , Mei Lu","doi":"10.1016/j.disc.2025.114802","DOIUrl":"10.1016/j.disc.2025.114802","url":null,"abstract":"<div><div>A graph <em>H</em> is said to be <em>F</em>-saturated relative to <em>G</em>, if <em>H</em> does not contain any copy of <em>F</em>, but the addition of any edge <em>e</em> in <span><math><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>﹨</mo><mi>E</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> would create a copy of <em>F</em>. The minimum size of an <em>F</em>-saturated graph relative to <em>G</em> is denoted by <span><math><mi>s</mi><mi>a</mi><mi>t</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span>. Let <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> be the complete <em>k</em>-partite graph containing <em>n</em> vertices in each part and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span> be the cycle of length <em>ℓ</em>. In this paper we give an asymptotically tight bound of <span><math><mi>s</mi><mi>a</mi><mi>t</mi><mo>(</mo><msubsup><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>n</mi></mrow></msubsup><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></math></span> for all <span><math><mi>ℓ</mi><mo>≥</mo><mn>4</mn><mo>,</mo><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> except <span><math><mo>(</mo><mi>ℓ</mi><mo>,</mo><mi>k</mi><mo>)</mo><mo>=</mo><mo>(</mo><mn>4</mn><mo>,</mo><mn>4</mn><mo>)</mo></math></span>. Moreover, we determine the exact value of <span><math><mi>s</mi><mi>a</mi><mi>t</mi><mo>(</mo><msubsup><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>n</mi></mrow></msubsup><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></math></span> for <span><math><mi>k</mi><mo>></mo><mi>ℓ</mi><mo>=</mo><mn>4</mn></math></span> and <span><math><mn>5</mn><mo>≥</mo><mi>ℓ</mi><mo>></mo><mi>k</mi><mo>≥</mo><mn>3</mn></math></span> and <span><math><mo>(</mo><mi>ℓ</mi><mo>,</mo><mi>k</mi><mo>)</mo><mo>=</mo><mo>(</mo><mn>6</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114802"},"PeriodicalIF":0.7,"publicationDate":"2025-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145109496","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 improved result on the stability of odd cycles","authors":"Zilong Yan , Xiaoli Yuan, Yuejian Peng","doi":"10.1016/j.disc.2025.114801","DOIUrl":"10.1016/j.disc.2025.114801","url":null,"abstract":"<div><div>Let <span><math><mi>C</mi></math></span> be a family consisting of some odd cycles. Suppose that <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>ℓ</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> is the shortest odd cycle not in <span><math><mi>C</mi></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> is the longest odd cycle in <span><math><mi>C</mi></math></span>. Let <span><math><mi>B</mi><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>ℓ</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> denote the graph obtained by taking <span><math><mn>2</mn><mi>ℓ</mi><mo>+</mo><mn>1</mn></math></span> vertex-disjoint copies of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn><mo>(</mo><mn>2</mn><mi>ℓ</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mfrac><mo>,</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn><mo>(</mo><mn>2</mn><mi>ℓ</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mfrac></mrow></msub></math></span> and selecting a vertex in each of them such that these vertices form a cycle of length <span><math><mn>2</mn><mi>ℓ</mi><mo>+</mo><mn>1</mn></math></span>. In this paper, we show that if <span><math><mi>k</mi><mo>≥</mo><msup><mrow><mn>79</mn></mrow><mrow><mn>4</mn></mrow></msup><msup><mrow><mi>ℓ</mi></mrow><mrow><mn>12</mn></mrow></msup></math></span>, <span><math><mi>n</mi><mo>≥</mo><mn>4</mn><mo>(</mo><mn>2</mn><mi>ℓ</mi><mo>+</mo><mn>1</mn><mo>)</mo><mi>k</mi><mo>+</mo><mo>(</mo><mn>16</mn><mi>ℓ</mi><mo>+</mo><mn>10</mn><mo>)</mo><mo>(</mo><mn>2</mn><mi>ℓ</mi><mo>+</mo><mn>1</mn><mo>)</mo><msup><mrow><mi>k</mi></mrow><mrow><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup></math></span> and <em>G</em> is an <em>n</em>-vertex <span><math><mi>C</mi></math></span>-free graph with minimum degree <span><math><mi>δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>></mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn><mo>(</mo><mn>2</mn><mi>ℓ</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mfrac></math></span>, then <em>G</em> is bipartite. The condition on the minimum degree is tight evidenced by <span><math><mi>B</mi><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>ℓ</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>. Furthermore, we show the only non-bipartite <span><math><mi>C</mi></math></span>-free graph with minimum degree <span><math><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn><mo>(</mo><mn>2</mn><mi>ℓ</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mfrac></math></span> is <span><math><mi>B</mi><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>ℓ</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>. This improves the condition of <em>n</em> in a result of Yuan-Peng. The previous known result of Yuan-Peng corresponding to the case <span><math><mi>C</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>C</mi></mrow><mr","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114801"},"PeriodicalIF":0.7,"publicationDate":"2025-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145109494","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":"Hamiltonicity of transitive graphs whose automorphism group has Zp as commutator subgroups","authors":"Florian Lehner , Farzad Maghsoudi , Babak Miraftab","doi":"10.1016/j.disc.2025.114798","DOIUrl":"10.1016/j.disc.2025.114798","url":null,"abstract":"<div><div>In 1982, Durnberger proved that every connected Cayley graph of a finite group with a commutator subgroup of prime order contains a hamiltonian cycle. In this paper, we extend this result to the infinite case. Additionally, we generalize this result to a broader class of infinite graphs <em>X</em>, where the automorphism group of <em>X</em> contains a transitive subgroup <em>G</em> with a cyclic commutator subgroup of prime order.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114798"},"PeriodicalIF":0.7,"publicationDate":"2025-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145110006","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 spectral conditions for ID-factor-criticality and fractional [a,b]-factor of graphs","authors":"Tingyan Ma , Ligong Wang","doi":"10.1016/j.disc.2025.114803","DOIUrl":"10.1016/j.disc.2025.114803","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>,</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span> be a graph with vertex set <span><math><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and edge set <span><math><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. A graph is <em>ID</em>-factor-critical if for every independent set <em>I</em> of <em>G</em> whose size has the same parity as <span><math><mo>|</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo></math></span>, <span><math><mi>G</mi><mo>−</mo><mi>I</mi></math></span> has a perfect matching. For two positive integers <em>a</em> and <em>b</em> with <span><math><mi>a</mi><mo>≤</mo><mi>b</mi></math></span>, let <em>h</em>: <span><math><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>→</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span> be a function on <span><math><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> satisfying <span><math><mi>a</mi><mo>≤</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>e</mi><mo>∈</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow></msub><mi>h</mi><mo>(</mo><mi>e</mi><mo>)</mo><mo>≤</mo><mi>b</mi></math></span> for any vertex <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. Then the spanning subgraph with edge set <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>h</mi></mrow></msub></math></span>, denoted by <span><math><mi>G</mi><mo>[</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>h</mi></mrow></msub><mo>]</mo></math></span>, is called a fractional <span><math><mo>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo></math></span>-factor of <em>G</em> with indicator function <em>h</em>, where <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>h</mi></mrow></msub><mo>=</mo><mo>{</mo><mi>e</mi><mo>∈</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mi>h</mi><mo>(</mo><mi>e</mi><mo>)</mo><mo>></mo><mn>0</mn><mo>}</mo></math></span> and <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo><mo>=</mo><mo>{</mo><mi>e</mi><mo>∈</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mi>e</mi></math></span> is incident with <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> in <em>G</em>}. A graph is defined as a fractional <span><math><mo>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo></math></span>-deleted graph if for any <span><math><mi>e</mi><mo>∈</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, <span><math><mi>G</mi><mo>−</mo><mi>e</mi></math></span> contains a fractional <span><math><mo>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo></math></span>-factor. For any integer <span><math><mi>k</mi><mo>≥</mo><mn>1</mn></math></span>, a graph has a <em>k<","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114803"},"PeriodicalIF":0.7,"publicationDate":"2025-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145109495","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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