Discrete Applied Mathematics最新文献

{"title":"Subprime and superprime graphs","authors":"Houmem Belkhechine ,&nbsp;Cherifa Ben Salha ,&nbsp;Rim Romdhane","doi":"10.1016/j.dam.2025.06.016","DOIUrl":"10.1016/j.dam.2025.06.016","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a graph with at least four vertices. The graph <span><math><mi>G</mi></math></span> is prime if all its modules are trivial. For example, for every integer <span><math><mrow><mi>n</mi><mo>≥</mo><mn>4</mn></mrow></math></span>, the path <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is prime. So the graph <span><math><mi>G</mi></math></span> can be made prime by modifying the adjacency relation between some pairs of vertices, i.e., by adding some edges to <span><math><mi>G</mi></math></span> and removing some other edges from it. (The first two authors proved that the graph <span><math><mi>G</mi></math></span> can be made prime by at most <span><math><mrow><mrow><mo>|</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>−</mo><mn>1</mn></mrow></math></span> adjacency modifications.) Nevertheless, there are graphs that cannot be made prime by only adding or only removing edges. So let us say that <span><math><mi>G</mi></math></span> is superprime (resp. subprime) if it can be made prime by removing (resp. adding) edges. Given that subprime graphs are the complements of superprime ones, we have chosen to focus solely on superprime graphs, which are graphs admitting a spanning prime subgraph. These graphs are connected. They were considered by D.P. Sumner, who proved that given a connected graph <span><math><mi>G</mi></math></span> with at least four vertices, <span><math><mi>G</mi></math></span> is superprime if it does not admit a stable module of size 2. This sufficient condition for <span><math><mi>G</mi></math></span> to be superprime is not necessary. In this paper, we provide a necessary and sufficient condition for <span><math><mi>G</mi></math></span> to be superprime. This condition involves neighborhood complexes that we associate with stable sets.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"376 ","pages":"Pages 133-140"},"PeriodicalIF":1.0,"publicationDate":"2025-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144366639","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":"Common matching number of a graph","authors":"Magda Dettlaff ,&nbsp;Magdalena Lemańska ,&nbsp;Jerzy Topp","doi":"10.1016/j.dam.2025.06.014","DOIUrl":"10.1016/j.dam.2025.06.014","url":null,"abstract":"&lt;div&gt;&lt;div&gt;The cardinality of the largest matching in a graph &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, denoted by &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, is referred to as the upper matching number of &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. The lower matching number &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is defined as the cardinality of the smallest maximal matching in &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. We introduce the concept of the common matching number of a graph &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, denoted by &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, which is the largest integer &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; such that every edge in &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; belongs to a matching that contains at least &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; edges. In this paper, we explore the relationships between the parameters &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. In particular, we demonstrate that the difference between &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; can be arbitrarily large, while the difference between &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; can at most be one. Additionally, we characterize the trees &lt;span&gt;&lt;math&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; for which &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, as well as the trees &lt;span&gt;&lt;math&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; for which &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"376 ","pages":"Pages 50-61"},"PeriodicalIF":1.0,"publicationDate":"2025-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144364675","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 extrema of F2-free graphs with given size revisited","authors":"Yuan Chen ,&nbsp;Shuchao Li ,&nbsp;Luying Zhang ,&nbsp;Minjie Zhang","doi":"10.1016/j.dam.2025.06.007","DOIUrl":"10.1016/j.dam.2025.06.007","url":null,"abstract":"&lt;div&gt;&lt;div&gt;A graph &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is said to be &lt;span&gt;&lt;math&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-free if it does not contain &lt;span&gt;&lt;math&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; as a subgraph. Let &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; be the friendship graph obtained from &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; triangles by sharing a common vertex. Recently, Li et al. (2023) showed that &lt;span&gt;&lt;span&gt;&lt;span&gt;(&lt;span&gt;&lt;math&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;)&lt;/span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msqrt&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msqrt&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;holds for all &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;-free graphs &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; of size &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, and equality in &lt;span&gt;&lt;span&gt;(&lt;span&gt;&lt;math&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;)&lt;/span&gt;&lt;/span&gt; holds if and only if &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; has exactly one non-trivial component &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∨&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∨&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; denotes the join of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. The bound in &lt;span&gt;&lt;span&gt;(&lt;span&gt;&lt;math&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;)&lt;/span&gt;&lt;/span&gt; is sharp only for odd &lt;span&gt;&lt;math&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and the extremal graph has some isolated vertices if the order &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. Denote by &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; the set of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;-free graphs with &lt;span&gt;&lt;math&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; edges having no isolated vertices. In this paper, as a continuous research of Li, Lu and Peng (2023), a sharp upper bound on the spectral radius of graphs among &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; for even &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;16&lt;/mn&gt;&lt;/mr","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"375 ","pages":"Pages 294-307"},"PeriodicalIF":1.0,"publicationDate":"2025-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144338920","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":"Perfect triple Roman domination","authors":"M. Kor ,&nbsp;J. Amjadi ,&nbsp;M. Chellali ,&nbsp;S.M. Sheikholeslami","doi":"10.1016/j.dam.2025.06.010","DOIUrl":"10.1016/j.dam.2025.06.010","url":null,"abstract":"<div><div>Let <span><math><mi>f</mi></math></span> be a function that assigns labels from the set <span><math><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>}</mo></mrow></math></span> to the vertices of a simple graph <span><math><mi>G</mi></math></span>. The active neighborhood <span><math><mrow><mi>A</mi><mi>N</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></math></span> of a vertex <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> with respect to <span><math><mi>f</mi></math></span> is the set of all neighbors of <span><math><mi>v</mi></math></span> that are assigned non-zero values under <span><math><mi>f</mi></math></span>. The function <span><math><mi>f</mi></math></span> is a perfect triple Roman dominating function (PTRD-function) on <span><math><mi>G</mi></math></span> if for every vertex <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>&lt;</mo><mn>3</mn></mrow></math></span>, we have <span><math><mrow><msub><mrow><mo>∑</mo></mrow><mrow><mi>u</mi><mo>∈</mo><mi>N</mi><mrow><mo>[</mo><mi>v</mi><mo>]</mo></mrow></mrow></msub><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>|</mo><mi>A</mi><mi>N</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>+</mo><mn>3</mn></mrow></math></span>. The weight of a PTRD-function is the sum of its function values over the whole set of vertices, and the PTRD-number is the minimum weight of a PTRD-function on <span><math><mi>G</mi></math></span>. In this paper, we show that determining the PTRD-number is NP-complete even when restricted to bipartite graphs. Moreover, the exact values of the PTRD-number for paths and cycles are established. Moreover, we provide an upper bound for the PTRD-number for trees of order at least five and we characterize the extremal trees attaining this upper bound.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"376 ","pages":"Pages 41-49"},"PeriodicalIF":1.0,"publicationDate":"2025-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144364607","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":"Extremal numbers of leaves for trees with fixed diameter and maximum degree","authors":"Xing Feng ,&nbsp;Zejun Huang","doi":"10.1016/j.dam.2025.06.005","DOIUrl":"10.1016/j.dam.2025.06.005","url":null,"abstract":"<div><div>In 1975, Lesniak established a lower bound on the number of leaves in trees with given order and diameter, which is sharp when the diameter is even. Qiao and Zhan later determined the exact minimum number of leaves in such trees. In this note, we determine both the minimum number and the maximum number of leaves for trees with given order, diameter and maximum degree.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"375 ","pages":"Pages 290-293"},"PeriodicalIF":1.0,"publicationDate":"2025-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144338226","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 existence of a {K1,2,K1,3,K5}-factor based on the size or the Aα-spectral radius of graphs","authors":"Xianglong Zhang,&nbsp;Lihua You","doi":"10.1016/j.dam.2025.06.003","DOIUrl":"10.1016/j.dam.2025.06.003","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; be a connected graph of order &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. A &lt;span&gt;&lt;math&gt;&lt;mrow&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;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&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;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&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;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;-factor of &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is a spanning subgraph of &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, in which each component is isomorphic to a member in &lt;span&gt;&lt;math&gt;&lt;mrow&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;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&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;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&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;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. The &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;-spectral radius of &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is denoted by &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. In this paper, we obtain a lower bound on the size (resp. the &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;-spectral radius for &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;) of &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; to guarantee that &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; has a &lt;span&gt;&lt;math&gt;&lt;mrow&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;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&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;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&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;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;-factor, and show that &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; has a &lt;span&gt;&lt;math&gt;&lt;mrow&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;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&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;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&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;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;-factor if &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mi&gt;τ&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;τ&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is the largest root of &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"376 ","pages":"Pages 22-30"},"PeriodicalIF":1.0,"publicationDate":"2025-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144364606","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":"Reconstruction of segmentationally distributive space from its metric","authors":"Jan Pavlík","doi":"10.1016/j.dam.2025.05.027","DOIUrl":"10.1016/j.dam.2025.05.027","url":null,"abstract":"<div><div>In this paper we present new insights on the metric on segmentationally distributive space with the main focus on the finite ones. The research is build on the results presented in Pavlík (2023) and focuses on the induced metric. The main message brought by the paper is that each SD-space can be uniquely reconstructed from its induced metric.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"376 ","pages":"Pages 88-111"},"PeriodicalIF":1.0,"publicationDate":"2025-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144366517","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 n-ary initial literal and literal shuffle","authors":"Stefan Hoffmann","doi":"10.1016/j.dam.2025.04.023","DOIUrl":"10.1016/j.dam.2025.04.023","url":null,"abstract":"<div><div>The literal and initial literal shuffle operations were introduced to model the synchronization of two processes. However, due to their non-associative nature, they are inadequate for modeling the synchronization of multiple processes. In this work, we extend both operations to <span><math><mi>n</mi></math></span>-ary versions, enabling the modeling of multiple synchronized processes. We also introduce “iterated” variants of these <span><math><mi>n</mi></math></span>-ary shuffles, which differ from previously proposed iterated versions for the two-argument cases. We investigate formal properties and demonstrate that in a full trio their expressive power coincides with that of the general shuffle operation. Furthermore, we examine closure properties and explore various decision problems inspired by analogous problems for the general shuffle operation. For instance, determining whether a word belongs to the <span><math><mi>n</mi></math></span>-ary initial literal shuffle of <span><math><mi>n</mi></math></span> words is a decision problem in <span>L</span>, whereas the corresponding problem for the <span><math><mi>n</mi></math></span>-ary literal shuffle is <span>NP</span>-complete.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"376 ","pages":"Pages 112-132"},"PeriodicalIF":1.0,"publicationDate":"2025-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144366638","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":"Ordered Gallai–Ramsey numbers","authors":"Yaping Mao","doi":"10.1016/j.dam.2025.06.018","DOIUrl":"10.1016/j.dam.2025.06.018","url":null,"abstract":"<div><div>An <em>ordered graph</em> is a pair <span><math><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>A</mi><mo>)</mo></mrow></math></span> where <span><math><mi>A</mi></math></span> is an ordering of <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> of a graph <span><math><mi>G</mi></math></span>. For given ordered graphs <span><math><mrow><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>A</mi><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mo>,</mo><mo>…</mo><mo>,</mo><mrow><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>, the <em>ordered Gallai–Ramsey number</em> <span><math><mrow><msub><mrow><mover><mrow><mi>g</mi><mi>r</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>A</mi><mo>)</mo></mrow><mo>:</mo><mrow><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mo>,</mo><mo>…</mo><mo>,</mo><mrow><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> is defined as the smallest number <span><math><mi>N</mi></math></span> such that every <span><math><mi>k</mi></math></span>-edge-coloring of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>N</mi></mrow></msub></math></span> contains an order preserving rainbow copy of <span><math><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>A</mi><mo>)</mo></mrow></math></span> or an order preserving monochromatic copy of <span><math><mrow><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow></math></span> with color <span><math><mi>i</mi></math></span> as an ordered subgraph. In this paper, we first studied the ordered complete graphs without small rainbow subgraphs, like stars, paths, and matchings. Next, we give the exact values or bounds for the ordered Gallai–Ramsey numbers.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"376 ","pages":"Pages 62-71"},"PeriodicalIF":1.0,"publicationDate":"2025-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144364674","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":"Gracefulness of two nested cycles: A first approach","authors":"Miguel Licona,&nbsp;Joaquín Tey","doi":"10.1016/j.dam.2025.06.008","DOIUrl":"10.1016/j.dam.2025.06.008","url":null,"abstract":"<div><div>It is known that if a plane graph admits a graceful (resp. near-graceful) labeling, then its semidual admits a conservative (resp. near-conservative) labeling. Consequently, we studied gracefulness of two nested cycles graphs considering two different perspectives: the first one by finding graceful (near-graceful) labelings of two nested cycles graphs, and the other one by finding conservative (near-conservative) labelings of its semidual. In this work we prove that for a given integer <span><math><mrow><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≥</mo><mn>3</mn></mrow></math></span>, there exists an integer <span><math><mrow><msup><mrow><mi>m</mi></mrow><mrow><mo>∗</mo></mrow></msup><mo>&gt;</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math></span> such that for all <span><math><mrow><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>≥</mo><msup><mrow><mi>m</mi></mrow><mrow><mo>∗</mo></mrow></msup></mrow></math></span>, if <span><math><mrow><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>≡</mo><mn>0</mn><mspace></mspace><mtext>or</mtext><mspace></mspace><mn>3</mn><mspace></mspace><mrow><mo>(</mo><mo>mod</mo><mspace></mspace><mn>4</mn><mo>)</mo></mrow></mrow></math></span> (resp. <span><math><mrow><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>≡</mo><mn>1</mn><mo>,</mo><mn>2</mn><mspace></mspace><mrow><mo>(</mo><mo>mod</mo><mspace></mspace><mn>4</mn><mo>)</mo></mrow></mrow></math></span>), then there exists a graceful (resp. near-graceful) plane cycle with chords consisting of two nested cycles of sizes <span><math><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, respectively. We also show that the semidual of a plane cycle with chords of size <span><math><mi>M</mi></math></span> consisting of two nested cycles is conservative if <span><math><mrow><mi>M</mi><mo>≡</mo><mn>0</mn><mspace></mspace><mtext>or</mtext><mspace></mspace><mn>3</mn><mspace></mspace><mrow><mo>(</mo><mo>mod</mo><mspace></mspace><mn>4</mn><mo>)</mo></mrow></mrow></math></span>, and it is near-conservative otherwise.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"376 ","pages":"Pages 72-87"},"PeriodicalIF":1.0,"publicationDate":"2025-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144364676","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}