Discrete Applied Mathematics最新文献

{"title":"A new locally t-diagnosable structure under the PMC model with an application to matching composition networks","authors":"Meirun Chen ,&nbsp;Cheng-Kuan Lin ,&nbsp;Kung-Jui Pai","doi":"10.1016/j.dam.2025.05.015","DOIUrl":"10.1016/j.dam.2025.05.015","url":null,"abstract":"<div><div>The PMC model is the test-based diagnosis in which a node performs the diagnosis by testing the neighbor nodes via the links between them. If we concentrate on the status of some nodes then instead of doing the global test, Hsu and Tan proposed the concept of local diagnosis and two structures to diagnose a node under the PMC model. To better evaluate the local diagnosability of a node, we propose a new structure and the related algorithm to diagnose a node under the PMC model in this paper. Applying the two structures proposed by Hsu and Tan, and the new structure we propose in this paper, we determine the accurate value of the local diagnosability of each node in matching composition networks. Simulation results are presented, showing the performance of our algorithm. It shows that even if the failure probability of a node is 0.4, our algorithm can still determine the state of a node with the accuracy above 0.9.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"374 ","pages":"Pages 1-15"},"PeriodicalIF":1.0,"publicationDate":"2025-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144069362","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":"Hybrid intermittent fault diagnosis of general graphs","authors":"Lulu Yang ,&nbsp;Shuming Zhou ,&nbsp;Weixing Zheng","doi":"10.1016/j.dam.2025.05.009","DOIUrl":"10.1016/j.dam.2025.05.009","url":null,"abstract":"<div><div>With the rapid development of information technology, networks have emerged as a crucial infrastructure in the big data era. System-level fault diagnosis plays a vital role to locate and repair faulty nodes in networks. However, the majority of research primarily focus on diagnosing faulty nodes of regular networks, with comparably less attention devoted to fault identification in irregular networks under the circumstance of link failures. In this paper, we introduce the notion of hybrid intermittent fault diagnosability and derive the corresponding diagnosability for general networks. Additionally, we determine the hybrid intermittent fault diagnosability for various well-known networks. Furthermore, we propose a HIFPD-MM* algorithm, which possesses a time complexity of <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>k</mi><mo>×</mo><mrow><mo>|</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo></mrow><mi>⋅</mi><msup><mrow><mrow><mo>(</mo><mi>Δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span>, where <span><math><mi>k</mi></math></span> denotes the number of stages of the algorithm in one round, and <span><math><mrow><mi>Δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> denotes the maximum degree of graph <span><math><mi>G</mi></math></span>. Through extensive experiments conducted on hypercubes and real-world datasets, we validate the effectiveness and accuracy of our proposed algorithm.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"374 ","pages":"Pages 16-32"},"PeriodicalIF":1.0,"publicationDate":"2025-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143948823","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":"Forbidden pairs for 2-factorable and hamiltonian graphs under the necessary condition","authors":"Qiang Wang ,&nbsp;Liming Xiong","doi":"10.1016/j.dam.2025.05.017","DOIUrl":"10.1016/j.dam.2025.05.017","url":null,"abstract":"<div><div>In <span><span>[1]</span></span>, Wang and Xiong characterize all forbidden pairs (not necessary connected) <span><math><mrow><mi>R</mi><mo>,</mo><mi>S</mi></mrow></math></span> such that 2-connected <span><math><mrow><mo>{</mo><mi>R</mi><mo>,</mo><mi>S</mi><mo>}</mo></mrow></math></span>-free graph <span><math><mi>G</mi></math></span> admitting a 2-factor is hamiltonian. To be more comprehensive, in this paper, we characterize all forbidden pairs (not necessary connected) <span><math><mrow><mi>R</mi><mo>,</mo><mi>S</mi></mrow></math></span> such that connected (or 2-edge-connected) <span><math><mrow><mo>{</mo><mi>R</mi><mo>,</mo><mi>S</mi><mo>}</mo></mrow></math></span>-free graph <span><math><mi>G</mi></math></span> admitting a 2-factor is hamiltonian. Besides, we characterize all forbidden pairs (not necessary connected) <span><math><mrow><mi>R</mi><mo>,</mo><mi>S</mi></mrow></math></span> such that connected (or 2-edge-connected) <span><math><mrow><mo>{</mo><mi>R</mi><mo>,</mo><mi>S</mi><mo>}</mo></mrow></math></span>-free graph <span><math><mi>G</mi></math></span> admitting an even-factor has a 2-factor. Comparing with the main result of Yang and Xiong (2023), we give all disconnected forbidden pairs. In the end, we find all forbidden pairs <span><math><mrow><mi>R</mi><mo>,</mo><mi>S</mi></mrow></math></span> such that connected (or 2-edge-connected) <span><math><mrow><mo>{</mo><mi>R</mi><mo>,</mo><mi>S</mi><mo>}</mo></mrow></math></span>-free graph <span><math><mi>G</mi></math></span> who has an even-factor is hamiltonian.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"373 ","pages":"Pages 290-300"},"PeriodicalIF":1.0,"publicationDate":"2025-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143947461","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 saturation game","authors":"Balázs Patkós ,&nbsp;Miloš Stojaković ,&nbsp;Jelena Stratijev ,&nbsp;Máté Vizer","doi":"10.1016/j.dam.2025.05.014","DOIUrl":"10.1016/j.dam.2025.05.014","url":null,"abstract":"<div><div>We study the following game version of the generalized graph Turán problem. For two fixed graphs <span><math><mi>F</mi></math></span> and <span><math><mi>H</mi></math></span>, two players, Max and Mini, alternately claim unclaimed edges of the complete graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> such that the graph <span><math><mi>G</mi></math></span> of the claimed edges must remain <span><math><mi>F</mi></math></span>-free throughout the game. The game ends when no further edges can be claimed, i.e. when <span><math><mi>G</mi></math></span> becomes <span><math><mi>F</mi></math></span>-saturated. The <span><math><mi>H</mi></math></span>-score of the game is the number of copies of <span><math><mi>H</mi></math></span> in <span><math><mi>G</mi></math></span>. Max aims to maximize the <span><math><mi>H</mi></math></span>-score, while Mini wants to minimize it. The <span><math><mi>H</mi></math></span>-score of the game when both players play optimally is denoted by <span><math><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>#</mi><mi>H</mi><mo>,</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> when Max starts, and by <span><math><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>#</mi><mi>H</mi><mo>,</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> when Mini starts. We study these values for several natural choices of <span><math><mi>F</mi></math></span> and <span><math><mi>H</mi></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"374 ","pages":"Pages 33-49"},"PeriodicalIF":1.0,"publicationDate":"2025-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144069361","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":"On the Nth 2-adic complexity of binary sequences identified with algebraic 2-adic integers","authors":"Zhixiong Chen ,&nbsp;Arne Winterhof","doi":"10.1016/j.dam.2025.05.024","DOIUrl":"10.1016/j.dam.2025.05.024","url":null,"abstract":"<div><div>We identify a binary sequence <span><math><mrow><mi>S</mi><mo>=</mo><msubsup><mrow><mrow><mo>(</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>∞</mi></mrow></msubsup></mrow></math></span> with the 2-adic integer <span><math><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow><mo>=</mo><munderover><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>∞</mi></mrow></munderover><msub><mrow><mi>s</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span>. In the case that <span><math><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span> is algebraic over <span><math><mi>Q</mi></math></span> of degree <span><math><mrow><mi>d</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, we prove that the <span><math><mi>N</mi></math></span>th 2-adic complexity of <span><math><mi>S</mi></math></span> is at least <span><math><mrow><mfrac><mrow><mi>N</mi></mrow><mrow><mi>d</mi></mrow></mfrac><mo>+</mo><mi>O</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>, where the implied constant depends only on the minimal polynomial of <span><math><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span>. This result is an analog of the bound of Mérai and the second author on the linear complexity of automatic sequences, that is, sequences with algebraic <span><math><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span> over the rational function field <span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span>.</div><div>We further discuss the most important case <span><math><mrow><mi>d</mi><mo>=</mo><mn>2</mn></mrow></math></span> in both settings and explain that the intersection of the set of 2-adic algebraic sequences and the set of automatic sequences is the set of (eventually) periodic sequences. Finally, we provide some experimental results supporting the conjecture that 2-adic algebraic sequences can have also a desirable <span><math><mi>N</mi></math></span>th linear complexity and automatic sequences a desirable <span><math><mi>N</mi></math></span>th 2-adic complexity, respectively.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"373 ","pages":"Pages 279-289"},"PeriodicalIF":1.0,"publicationDate":"2025-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143947561","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":"Unavoidable patterns in 2-colorings of the complete bipartite graph","authors":"Adriana Hansberg ,&nbsp;Denae Ventura","doi":"10.1016/j.dam.2025.05.018","DOIUrl":"10.1016/j.dam.2025.05.018","url":null,"abstract":"<div><div>We determine the colored patterns that appear in any 2-edge coloring of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span>, with <span><math><mi>n</mi></math></span> large enough and with sufficient edges in each color. We prove the existence of a positive integer <span><math><msub><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> such that any 2-edge coloring of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span> with at least <span><math><msub><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> edges in each color contains at least one of these patterns. We give a general upper bound for <span><math><msub><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and prove its tightness for some cases. We define the concepts of bipartite <span><math><mi>r</mi></math></span>-tonality and bipartite omnitonality using the complete bipartite graph as a base graph. We provide a characterization for bipartite <span><math><mi>r</mi></math></span>-tonal graphs and prove that every tree is bipartite omnitonal. Finally, we define the bipartite-balancing number and provide the exact bipartite-balancing number for paths and stars.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"374 ","pages":"Pages 50-60"},"PeriodicalIF":1.0,"publicationDate":"2025-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144069363","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":"Conditional matroidal edge connectivity of Cayley graphs","authors":"Li Wang ,&nbsp;Mingxi Su ,&nbsp;Liqing Lin ,&nbsp;Xiaohui Hua","doi":"10.1016/j.dam.2025.05.011","DOIUrl":"10.1016/j.dam.2025.05.011","url":null,"abstract":"<div><div>Edge connectivity is an important indicator to evaluate edge fault tolerance. As for a host of networks, the edge connectivity is exactly equal to their minimum degree. Conditional matroidal edge connectivity is a new graph edge connectivity parameter that can be defined when a partition of the edge set is given. It is a good indicator to restrict the different kinds of faulty edges. However, the determination of conditional matroidal edge connectivity is often limited by the edge transitivity of the corresponding graph. In this paper, we use the algebraic technique to conquer the limitation of edge transitivity and give a new method to calculate the conditional matroidal edge connectivity of Cayley graphs. As an application, we not only show the conditional matroidal edge connectivity of varietal hypercube <span><math><mrow><mi>V</mi><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span>, but also verify the conditional matroidal edge connectivity of alternating group graph <span><math><mrow><mi>A</mi><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span> that was studied by Zhang et al. (2022).</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"373 ","pages":"Pages 271-278"},"PeriodicalIF":1.0,"publicationDate":"2025-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143947560","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":"Use of simple arithmetic operations to construct efficiently implementable Boolean functions possessing high nonlinearity and good resistance to algebraic attacks","authors":"Claude Carlet ,&nbsp;Palash Sarkar","doi":"10.1016/j.dam.2025.05.004","DOIUrl":"10.1016/j.dam.2025.05.004","url":null,"abstract":"<div><div>We describe a new class of Boolean functions which provide the presently best known trade-off between low computational complexity, nonlinearity and (fast) algebraic immunity. In particular, for <span><math><mrow><mi>n</mi><mo>≤</mo><mn>20</mn></mrow></math></span>, we show that there are functions in the family achieving a combination of nonlinearity and (fast) algebraic immunity which is superior to what is achieved by any other efficiently implementable function. The main novelty of our approach is to apply a judicious combination of simple integer and binary field arithmetic to Boolean function construction.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"373 ","pages":"Pages 256-270"},"PeriodicalIF":1.0,"publicationDate":"2025-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143943126","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":"r-dynamic colorings and the spectral radius in graphs","authors":"Jiangdong Ai ,&nbsp;Suil O. ,&nbsp;Liwen Zhang","doi":"10.1016/j.dam.2025.05.003","DOIUrl":"10.1016/j.dam.2025.05.003","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;χ&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;mi&gt;ρ&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; be the chromatic number and spectral radius of &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, respectively. In 1967, Wilf proved that for a graph &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, we have &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;χ&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;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;ρ&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;. An &lt;span&gt;&lt;math&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-dynamic &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-coloring of a graph &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is a proper &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-coloring of &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; such that every vertex &lt;span&gt;&lt;math&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; in &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;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; has neighbors in at least &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;min&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; different color classes. The &lt;span&gt;&lt;math&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-dynamic chromatic number of a graph &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, written &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;χ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&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 the least &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; such that &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; has such a &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-coloring. Note that &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;χ&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;msub&gt;&lt;mrow&gt;&lt;mi&gt;χ&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;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;χ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&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;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mi&gt;Δ&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; (*) (Jahanbekama et al., 2016). By the inequality (*), we observe that for a positive integer &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and a connected graph &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, we have &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;χ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&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;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&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;mo&gt;.&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;&lt;/div&gt;&lt;div&gt;In this paper, for a positive integer &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, we provide graphs &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; with &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;χ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;m","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"373 ","pages":"Pages 249-255"},"PeriodicalIF":1.0,"publicationDate":"2025-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143935843","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":"Relation between the H-rank of a mixed graph and the girth of its underlying graph","authors":"Suliman Khan","doi":"10.1016/j.dam.2025.05.006","DOIUrl":"10.1016/j.dam.2025.05.006","url":null,"abstract":"<div><div>Let <span><math><mrow><msup><mrow><mi>Σ</mi></mrow><mrow><mi>π</mi></mrow></msup><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mrow><mo>(</mo><msup><mrow><mi>Σ</mi></mrow><mrow><mi>π</mi></mrow></msup><mo>)</mo></mrow><mo>,</mo><mi>E</mi><mrow><mo>(</mo><msup><mrow><mi>Σ</mi></mrow><mrow><mi>π</mi></mrow></msup><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> be a mixed graph obtained from a simple graph <span><math><mi>Γ</mi></math></span> with the same vertex set <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>Γ</mi><mo>)</mo></mrow></mrow></math></span> and an edge set <span><math><mrow><mi>E</mi><mrow><mo>(</mo><mi>Γ</mi><mo>)</mo></mrow></mrow></math></span> containing undirected edges and arcs. Let <span><math><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>Σ</mi></mrow><mrow><mi>π</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> be the (first kind of) Hermitian adjacency matrix of <span><math><msup><mrow><mi>Σ</mi></mrow><mrow><mi>π</mi></mrow></msup></math></span>. The <span><math><mi>H</mi></math></span>-rank of <span><math><msup><mrow><mi>Σ</mi></mrow><mrow><mi>π</mi></mrow></msup></math></span> is the rank of <span><math><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>Σ</mi></mrow><mrow><mi>π</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>, denoted by <span><math><mrow><msup><mrow><mi>r</mi></mrow><mrow><mi>H</mi></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>Σ</mi></mrow><mrow><mi>π</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>. The girth of <span><math><mi>Γ</mi></math></span> is the length of the shortest cycle in <span><math><mi>Γ</mi></math></span>, dented by <span><math><mrow><mi>g</mi><mrow><mo>(</mo><mi>Γ</mi><mo>)</mo></mrow></mrow></math></span> (or simply by <span><math><mi>g</mi></math></span>). In this paper, we show that under some conditions the <span><math><mi>H</mi></math></span>-rank of a mixed graph is equal to the girth of its underlying graph. Moreover, we characterize mixed graphs with <span><math><mi>H</mi></math></span>-rank <span><math><mrow><mi>g</mi><mo>−</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>g</mi><mo>+</mo><mn>2</mn></mrow></math></span>, distinct from the characterization of <span><math><mi>T</mi></math></span>-gain graphs provided by Khan (2024).</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"373 ","pages":"Pages 239-248"},"PeriodicalIF":1.0,"publicationDate":"2025-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143943125","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}