Discrete Applied Mathematics最新文献

{"title":"Oriented Hamiltonian paths in digraphs with arbitrary chromatic number","authors":"Ayman El Zein","doi":"10.1016/j.dam.2025.04.050","DOIUrl":"10.1016/j.dam.2025.04.050","url":null,"abstract":"<div><div>The Roy–Gallai theorem states that every <span><math><mi>n</mi></math></span>-chromatic digraph contains a directed path of order <span><math><mi>n</mi></math></span>. Several results have provided necessary conditions concerning the chromatic number to guarantee the existence of paths in digraphs. In this paper, we prove the existence of digraphs with an arbitrary chromatic number <span><math><mrow><mi>n</mi><mo>≥</mo><mn>8</mn></mrow></math></span> containing every oriented Hamiltonian path.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"372 ","pages":"Pages 295-297"},"PeriodicalIF":1.0,"publicationDate":"2025-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143902535","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":"Partitioning vertices of graphs into paths of the same length","authors":"Oleg Duginov ,&nbsp;Dmitriy Malyshev ,&nbsp;Dmitriy Mokeev","doi":"10.1016/j.dam.2025.04.054","DOIUrl":"10.1016/j.dam.2025.04.054","url":null,"abstract":"<div><div>Given a graph, the (<em>induced</em>) <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span><em>-partition problem</em> is to decide whether its vertex set can be partitioned into subsets, each of which induces (the <span><math><mi>k</mi></math></span>-path) a <span><math><mi>k</mi></math></span>-vertex subgraph with a Hamiltonian path. We show that these problems are NP-complete for planar subcubic bipartite <span><math><mrow><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></mrow></math></span>-free graphs of girth <span><math><mi>g</mi></math></span>, for any <span><math><mrow><mi>k</mi><mo>,</mo><mi>g</mi><mo>≥</mo><mn>3</mn><mo>,</mo><mi>l</mi><mo>≥</mo><mn>1</mn></mrow></math></span>, where <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is obtained by joining central vertices in two copies of <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> with <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>. We show that the <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>-partition (induced <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>-partition) problem is NP-complete for split graphs and any <span><math><mrow><mi>k</mi><mo>≥</mo><mn>5</mn></mrow></math></span>, chordal graphs and any <span><math><mrow><mi>k</mi><mo>≥</mo><mn>4</mn></mrow></math></span> (any <span><math><mrow><mi>k</mi><mo>≥</mo><mn>3</mn></mrow></math></span>), line graphs of planar bipartite graphs and any <span><math><mrow><mi>k</mi><mo>≥</mo><mn>5</mn></mrow></math></span> (any <span><math><mrow><mi>k</mi><mo>≥</mo><mn>3</mn></mrow></math></span>). We show that the <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-partition and, for any <span><math><mrow><mi>k</mi><mo>≥</mo><mn>5</mn></mrow></math></span>, induced <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>-partition problems, restricted to split graphs, are polynomial. Additionally, we prove NP-completeness for the optimization version of the induced <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-partition problem and split graphs.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"373 ","pages":"Pages 179-195"},"PeriodicalIF":1.0,"publicationDate":"2025-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143899295","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":"k-path-connectivity of the complete balanced tripartite graph Kn,n,n for n+1≤k≤2n−4","authors":"Shasha Li,&nbsp;Xiaoxue Gao,&nbsp;Qihui Jin","doi":"10.1016/j.dam.2025.04.043","DOIUrl":"10.1016/j.dam.2025.04.043","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Given a graph &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and a set &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;⊆&lt;/mo&gt;&lt;mi&gt;V&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;/mrow&gt;&lt;/math&gt;&lt;/span&gt; of size at least 2, an &lt;span&gt;&lt;math&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;em&gt;-path&lt;/em&gt; in &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is a path that connects all vertices of &lt;span&gt;&lt;math&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. Two &lt;span&gt;&lt;math&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-paths &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&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; are said to be &lt;em&gt;internally disjoint&lt;/em&gt; if &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;∩&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&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;mo&gt;=&lt;/mo&gt;&lt;mo&gt;0̸&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;∩&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&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;mo&gt;=&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. Let &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&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;/math&gt;&lt;/span&gt; denote the maximum number of internally disjoint &lt;span&gt;&lt;math&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-paths in &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. The &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;em&gt;-path-connectivity&lt;/em&gt; of &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;msub&gt;&lt;mrow&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&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;, is then defined as &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&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;=&lt;/mo&gt;&lt;mi&gt;min&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&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;mo&gt;|&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;⊆&lt;/mo&gt;&lt;mi&gt;V&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;mi&gt;a&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&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;mn&gt;2&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. Therefore, &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&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; is exactly the classical connectivity &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;κ&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;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&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; is exactly the maximum number of edge-disjoint Hamilton paths in &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. It is established that for &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"372 ","pages":"Pages 279-294"},"PeriodicalIF":1.0,"publicationDate":"2025-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143899433","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":"Exploring graphs with distinct M-eigenvalues: Product operation, Wronskian vertices, and controllability","authors":"Haiying Shan,&nbsp;Xiaoqi Liu","doi":"10.1016/j.dam.2025.04.055","DOIUrl":"10.1016/j.dam.2025.04.055","url":null,"abstract":"<div><div>Let <span><math><msup><mrow><mi>G</mi></mrow><mrow><mi>M</mi></mrow></msup></math></span> denote the set of connected graphs with distinct <span><math><mi>M</mi></math></span>-eigenvalues. This paper explores the <span><math><mi>M</mi></math></span>-spectrum and eigenvectors of a new product <span><math><mrow><mi>G</mi><msub><mrow><mo>∘</mo></mrow><mrow><mi>C</mi></mrow></msub><mi>H</mi></mrow></math></span> of graphs <span><math><mi>G</mi></math></span> and <span><math><mi>H</mi></math></span>. We present the necessary and sufficient condition for <span><math><mrow><mi>G</mi><msub><mrow><mo>∘</mo></mrow><mrow><mi>C</mi></mrow></msub><mi>H</mi></mrow></math></span> to have distinct <span><math><mi>M</mi></math></span>-eigenvalues. Specifically, for the rooted product <span><math><mrow><mi>G</mi><mo>∘</mo><mi>H</mi></mrow></math></span>, we present a more concise and precise condition. A key concept, <span><math><mi>M</mi></math></span>-Wronskian vertex, which plays a crucial role in determining graph properties related to separability and construction of specific graph families, is investigated. We propose a novel method for constructing infinite pairs of non-isomorphic <span><math><mi>M</mi></math></span>-cospectral graphs in <span><math><msup><mrow><mi>G</mi></mrow><mrow><mi>M</mi></mrow></msup></math></span> by leveraging the structural properties of <span><math><mi>M</mi></math></span>-Wronskian vertex. Moreover, the necessary and sufficient condition for <span><math><mrow><mi>G</mi><mo>∘</mo><mi>H</mi></mrow></math></span> to be <span><math><mi>M</mi></math></span>-controllable is given.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"373 ","pages":"Pages 125-136"},"PeriodicalIF":1.0,"publicationDate":"2025-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143890818","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":"Weighted Moon-type formulae for complete graphs and complete bipartite graphs","authors":"Jun Ge","doi":"10.1016/j.dam.2025.04.058","DOIUrl":"10.1016/j.dam.2025.04.058","url":null,"abstract":"<div><div>In 1964, Cayley’s formula was generalized to a fascinating formula on the number of spanning trees of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> containing a given spanning forest by J.W. Moon. It was not until 58 years later that the second Moon-type formula was discovered for <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span> by Dong and the author. In this note, we obtain weighted versions for these two Moon-type formulae with much shorter proofs.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"372 ","pages":"Pages 274-278"},"PeriodicalIF":1.0,"publicationDate":"2025-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143892255","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":"Queue layouts on folded hypercubes","authors":"Xin Geng,&nbsp;Yueyang Hao,&nbsp;Weihua Yang","doi":"10.1016/j.dam.2025.04.056","DOIUrl":"10.1016/j.dam.2025.04.056","url":null,"abstract":"<div><div>A queue layout of a graph <span><math><mi>G</mi></math></span> consists of a linear order of its vertices, and a partition of its edges into queues, such that no two edges in the same queue are nested. The queue number <span><math><mrow><mi>q</mi><mi>n</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is the minimum number of queues required in a queue layout of <span><math><mi>G</mi></math></span>. A graph <span><math><mi>G</mi></math></span> is a <span><math><mi>q</mi></math></span>-queue graph if <span><math><mrow><mi>q</mi><mi>n</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mi>q</mi></mrow></math></span>. Gregor et al. in Gregor et al. (2012) showed that the <span><math><mi>n</mi></math></span>-dimensional hypercube <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> has a layout into <span><math><mrow><mi>n</mi><mo>−</mo><mrow><mo>⌊</mo><mrow><msub><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></msub><mi>n</mi></mrow><mo>⌋</mo></mrow></mrow></math></span> queues for all <span><math><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></math></span>. Pai et al. in Pai et al. (0000) gave several upper bounds when <span><math><mrow><mi>n</mi><mo>≤</mo><mn>7</mn></mrow></math></span>. In particular, <span><math><mrow><mi>q</mi><mi>n</mi><mrow><mo>(</mo><mi>F</mi><msub><mrow><mi>Q</mi></mrow><mrow><mn>7</mn></mrow></msub><mo>)</mo></mrow><mo>≤</mo><mn>12</mn></mrow></math></span>. In this work, we generally obtain that <span><math><mrow><mi>F</mi><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span> has queue number at most <span><math><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></mrow></math></span> for all <span><math><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"373 ","pages":"Pages 154-158"},"PeriodicalIF":1.0,"publicationDate":"2025-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143890820","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":"Optimization tools for computing colorings of [1,…,n] with few monochromatic solutions on 3-variable linear equations","authors":"Jesús A. De Loera ,&nbsp;Denae Ventura ,&nbsp;Liuyue Wang ,&nbsp;William J. Wesley","doi":"10.1016/j.dam.2025.04.052","DOIUrl":"10.1016/j.dam.2025.04.052","url":null,"abstract":"<div><div>A famous result in arithmetic Ramsey theory says that for many linear homogeneous equations <span><math><mi>E</mi></math></span> there is a threshold value <span><math><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span> (the Rado number of <span><math><mi>E</mi></math></span>) such that for any <span><math><mi>k</mi></math></span>-coloring of the integers in the interval <span><math><mrow><mo>[</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>]</mo></mrow></math></span>, with <span><math><mrow><mi>n</mi><mo>≥</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span>, there exists at least one monochromatic solution. But one can further ask, <em>how many monochromatic solutions is the minimum possible in terms of</em> <span><math><mi>n</mi></math></span><em>?</em> Several authors have estimated this function before, here we offer new tools from integer and semidefinite optimization that help find either optimal or near optimal 2-colorings minimizing the number of monochromatic solutions of several families of 3-variable non-regular homogeneous linear equations. In the last part of the paper we further extend to three and more colors for the Schur equation, improving earlier work.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"373 ","pages":"Pages 159-178"},"PeriodicalIF":1.0,"publicationDate":"2025-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143890819","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":"Higher order matching preclusion for regular interconnection networks","authors":"Eddie Cheng,&nbsp;László Lipták,&nbsp;Lucian Mazza","doi":"10.1016/j.dam.2025.04.042","DOIUrl":"10.1016/j.dam.2025.04.042","url":null,"abstract":"<div><div>For a graph with an even number of vertices, the <em>matching preclusion number</em> is the minimum number of edges whose deletion results in a graph with no perfect matchings. The <em>conditional matching preclusion number</em>, introduced as an extension of the matching preclusion number, has the additional requirement that the resulting graph has no isolated vertices. In this paper we consider results related to a further generalization of this concept, called level 2 matching preclusion. We find sufficient conditions for a graph with girth at least 4 to be level 2 maximally matched and level 2 super matched. We apply these results to the class of pancake graphs, and show that they are level 2 maximally and super matched.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"373 ","pages":"Pages 107-124"},"PeriodicalIF":1.0,"publicationDate":"2025-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143879441","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":"Corrigendum to “Transit functions and pyramid-like binary clustering systems” [Discrete Appl. Math. 357 (2024) 365–384]","authors":"Manoj Changat ,&nbsp;Ameera Vaheeda Shanavas ,&nbsp;Peter F. Stadler","doi":"10.1016/j.dam.2025.04.041","DOIUrl":"10.1016/j.dam.2025.04.041","url":null,"abstract":"","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"371 ","pages":"Page 280"},"PeriodicalIF":1.0,"publicationDate":"2025-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143936079","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":"Completely independent spanning trees in kth power of 2-connected graphs","authors":"Xia Hong ,&nbsp;Zhizheng Zhang","doi":"10.1016/j.dam.2025.04.051","DOIUrl":"10.1016/j.dam.2025.04.051","url":null,"abstract":"<div><div>Let <span><math><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></math></span> be spanning trees of a graph <span><math><mi>G</mi></math></span>. For any two vertices <span><math><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></math></span> of <span><math><mi>G</mi></math></span>, if the paths from <span><math><mi>u</mi></math></span> to <span><math><mi>v</mi></math></span> in these <span><math><mi>k</mi></math></span> trees are pairwise openly disjoint, then we say that <span><math><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></math></span> are completely independent. Let <span><math><mrow><mi>G</mi><mo>≇</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span>, where <span><math><mrow><mi>n</mi><mo>=</mo><mi>q</mi><mi>k</mi><mo>+</mo><mn>2</mn><mo>,</mo><mi>k</mi><mo>≥</mo><mn>6</mn><mo>,</mo><mi>q</mi><mo>≥</mo><mn>3</mn></mrow></math></span>. In this paper, we prove that the <span><math><mi>k</mi></math></span>th power of any 2-connected graph <span><math><mi>G</mi></math></span> on <span><math><mrow><mi>n</mi><mo>≥</mo><mn>2</mn><mi>k</mi><mrow><mo>(</mo><mi>k</mi><mo>≥</mo><mn>4</mn><mo>)</mo></mrow></mrow></math></span> vertices has <span><math><mi>k</mi></math></span> completely independent spanning trees, unless <span><math><mrow><mi>G</mi><mo>≅</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span> where <span><math><mrow><mi>n</mi><mo>=</mo><mi>k</mi><mi>q</mi><mo>+</mo><mi>r</mi><mo>,</mo><mi>r</mi><mo>∈</mo><mrow><mo>{</mo><mn>3</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>}</mo></mrow></mrow></math></span> or <span><math><mrow><mi>n</mi><mo>=</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>2</mn><mo>,</mo><mi>k</mi><mo>≥</mo><mn>6</mn></mrow></math></span>. The work of this paper improves the results of Hong (2018).</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"372 ","pages":"Pages 268-273"},"PeriodicalIF":1.0,"publicationDate":"2025-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143874386","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}