Discrete Applied Mathematics最新文献

{"title":"Counting the minimum number of arcs in an oriented graph having weak diameter 2","authors":"Sandip Das ,&nbsp;Koushik Kumar Dey ,&nbsp;Pavan P.D. ,&nbsp;Sagnik Sen","doi":"10.1016/j.dam.2024.12.018","DOIUrl":"10.1016/j.dam.2024.12.018","url":null,"abstract":"<div><div>An oriented graph has weak diameter at most <span><math><mi>d</mi></math></span> if every non-adjacent pair of vertices are connected by a directed <span><math><mi>d</mi></math></span>-path. The function <span><math><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>d</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> denotes the minimum number of arcs in an oriented graph on <span><math><mi>n</mi></math></span> vertices having weak diameter <span><math><mi>d</mi></math></span>. Finding the exact value of <span><math><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>d</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> is a challenging problem even for <span><math><mrow><mi>d</mi><mo>=</mo><mn>2</mn></mrow></math></span>. This function was introduced by Katona and Szemeŕedi (1967), and after that several attempts were made to find its exact value by Znam (1970), Dawes and Meijer (1987), Füredi, Horak, Pareek and Zhu (1998), and Kostochka, Luczak, Simonyi and Sopena (1999) through improving its best known bounds. In that process, it was proved that this function is asymptotically equal to <span><math><mrow><mi>n</mi><msub><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></msub><mi>n</mi></mrow></math></span> and hence, is an asymptotically increasing function. However, the exact value and behavior of this function was not known.</div><div>In this article, we observe that the oriented graphs with weak diameter at most 2 are precisely the absolute oriented cliques, that is, analogues of cliques for oriented graphs in the context of oriented coloring. Through studying arc-minimal absolute oriented cliques we prove that <span><math><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> is a strictly increasing function. Furthermore, we improve the best known upper bound of <span><math><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> and conjecture that our upper bound is tight. This improvement of the upper bound improves known bounds involving the oriented achromatic number.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"364 ","pages":"Pages 222-236"},"PeriodicalIF":1.0,"publicationDate":"2024-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143142515","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 generalized Turán number of star forests","authors":"Yan-Jiao Liu,&nbsp;Jian-Hua Yin","doi":"10.1016/j.dam.2024.12.024","DOIUrl":"10.1016/j.dam.2024.12.024","url":null,"abstract":"&lt;div&gt;&lt;div&gt;The generalized Turán number ex&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&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;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is defined to be the maximum number of copies of a complete graph &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;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; in any &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-free graph on &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; vertices. Let &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; denote the star on &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; vertices, and let &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; denote the disjoint union of &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; copies of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;. Gan et al. (2015) and Chase (2020) determined ex&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&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;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; for &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. In this paper, we consider to investigate the generalized Turán number of star forests. We determine ex&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&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;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&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;/msub&gt;&lt;mo&gt;∪&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&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;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; for &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&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;msub&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&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;mn&gt;2&lt;/mn&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&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;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, and ex&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&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;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&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;/msub&gt;&lt;mo&gt;∪&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&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;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; for &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&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;mn&gt;2&lt;/mn&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&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;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;3&lt;/","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"364 ","pages":"Pages 213-221"},"PeriodicalIF":1.0,"publicationDate":"2024-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143142516","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":"A class of graphs that complementation makes infinitely many Cohen–Macaulay members","authors":"T. Ashitha ,&nbsp;T. Asir ,&nbsp;M.R. Pournaki","doi":"10.1016/j.dam.2024.12.016","DOIUrl":"10.1016/j.dam.2024.12.016","url":null,"abstract":"<div><div>Let <span><math><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></math></span> be integers greater than one and set <span><math><mrow><mrow><mo>[</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>]</mo></mrow><mo>=</mo><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>−</mo><mn>1</mn><mo>}</mo></mrow></mrow></math></span>, <span><math><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k</mi></mrow></math></span>. The graph <span><math><mrow><mi>G</mi><mrow><mo>(</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> is obtained by letting all the elements of <span><math><mrow><mrow><mo>[</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>]</mo></mrow><mo>×</mo><mo>⋯</mo><mo>×</mo><mrow><mo>[</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>]</mo></mrow></mrow></math></span> to be the vertices and defining distinct vertices <span><math><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow></math></span> and <span><math><mrow><mo>(</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow></math></span> to be adjacent if and only if <span><math><mrow><mo>gcd</mo><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span> for all <span><math><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k</mi></mrow></math></span>. In this paper, we show that this large class of graphs has just one Cohen–Macaulay member, namely <span><math><mrow><mi>G</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span>, and complementation makes infinitely many Cohen–Macaulay graphs.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"364 ","pages":"Pages 189-198"},"PeriodicalIF":1.0,"publicationDate":"2024-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143143038","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":"Fast computation of linear approximation of general word-oriented composite function","authors":"Sudong Ma ,&nbsp;Chenhui Jin ,&nbsp;Jie Guan ,&nbsp;Ziyu Guan ,&nbsp;Shuai Liu","doi":"10.1016/j.dam.2024.12.022","DOIUrl":"10.1016/j.dam.2024.12.022","url":null,"abstract":"&lt;div&gt;&lt;div&gt;The nonlinear component of word-oriented stream ciphers usually contains a Finite State Machine (FSM). The establishment of linear approximations of word-oriented FSM with high correlations is the basis of linear attack on stream cipher, including linear distinguishing attack and fast correlation attack. However, the existing methods can only give efficient algorithms for several specific word-oriented composite functions. In order to solve this problem, we first define a wider class of composite functions, namely Pseudo-Linear S-box Function Modulo &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; (PLSFM). New PLSFM extends the definition of Pseudo-Linear Function Modulo &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; (PLFM) by introducing S-box functions into PLFM, and covers more composite functions. Secondly, an efficient algorithm for fast calculating the correlation of a given linear approximation of PLSFM is proposed, which allows us to fast search for linear approximations with high correlations. Thirdly, we study the properties of linear approximations of a class of composite functions containing S-box functions, addition modulo &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; and subtraction modulo &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;. Finally, we give the linear approximations of MASHA stream cipher with absolute correlations of &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;23&lt;/mn&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;mn&gt;38&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; for the first time. If the controlled nonlinear feedback shift register of MASHA degenerates into a linear feedback function, a fast correlation attack with time/data/memory complexity of &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;197&lt;/mn&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;mn&gt;26&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;O&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;194&lt;/mn&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;mn&gt;96&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;O&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;196&lt;/mn&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;mn&gt;96&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; can be given. For SNOW 2.0 stream cipher, we find two other linear approximations with the current best correlation of &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;14&lt;/mn&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;mn&gt;41&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;, then we can give an improved fast correlation attack with time/data/memory complexity of &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;162&lt;/mn&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;mn&gt;91&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;O&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;161&lt;/mn&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;mn&gt;47&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;O&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;162&lt;/mn&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;mn&gt;47&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"364 ","pages":"Pages 157-172"},"PeriodicalIF":1.0,"publicationDate":"2024-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143142517","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":"Dual bounded generation: Polynomial, second-order cone and positive semidefinite matrix inequalities","authors":"Khaled Elbassioni","doi":"10.1016/j.dam.2024.12.020","DOIUrl":"10.1016/j.dam.2024.12.020","url":null,"abstract":"<div><div>In the monotone integer dualization problem, we are given two sets of vectors in an integer box such that no vector in the first set is dominated by a vector in the second. The question is to check if the two sets of vectors cover the entire integer box by upward and downward domination, respectively. It is known that the problem is (quasi-)polynomially equivalent to that of enumerating all maximal feasible solutions of a given monotone system of linear/separable/supermodular inequalities over integer vectors. The equivalence is established via showing that the dual family of minimal infeasible vectors has size bounded by a (quasi-)polynomial in the sizes of the family to be generated and the input description. Continuing in this line of work, in this paper, we consider systems of polynomial, second-order cone, and semidefinite inequalities. We give sufficient conditions under which such bounds can be established and highlight some applications.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"364 ","pages":"Pages 173-188"},"PeriodicalIF":1.0,"publicationDate":"2024-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143143043","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Thin edges in the subgraph induced by noncubic vertices of a brace","authors":"Xiaoling He,&nbsp;Fuliang Lu","doi":"10.1016/j.dam.2024.12.019","DOIUrl":"10.1016/j.dam.2024.12.019","url":null,"abstract":"<div><div>For a vertex set <span><math><mi>X</mi></math></span> in a graph, an <em>edge cut</em> <span><math><mrow><mi>∂</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span> is the set of edges with exactly one end vertex in <span><math><mi>X</mi></math></span>. An edge cut <span><math><mrow><mi>∂</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span> is <em>tight</em> if every perfect matching contains exactly one edge in <span><math><mrow><mi>∂</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span>. A matching covered bipartite graph is a <em>brace</em> if, for every tight cut <span><math><mrow><mi>∂</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mrow><mo>|</mo><mi>X</mi><mo>|</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span> or <span><math><mrow><mrow><mo>|</mo><mover><mrow><mi>X</mi></mrow><mo>¯</mo></mover><mo>|</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span>, where <span><math><mrow><mover><mrow><mi>X</mi></mrow><mo>¯</mo></mover><mo>=</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>∖</mo><mi>X</mi></mrow></math></span>. As one of the building blocks, braces play an important role in Lovász’s tight cut decomposition of matching covered graphs. An edge <span><math><mi>e</mi></math></span> in a brace <span><math><mi>G</mi></math></span> is <em>thin</em> if, for every tight cut <span><math><mrow><mi>∂</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span> of <span><math><mrow><mi>G</mi><mo>−</mo><mi>e</mi></mrow></math></span>, <span><math><mrow><mrow><mo>|</mo><mi>X</mi><mo>|</mo></mrow><mo>≤</mo><mn>3</mn></mrow></math></span> or <span><math><mrow><mrow><mo>|</mo><mover><mrow><mi>X</mi></mrow><mo>¯</mo></mover><mo>|</mo></mrow><mo>≤</mo><mn>3</mn></mrow></math></span>. Carvalho, Lucchesi and Murty conjectured that there exists a positive constant <span><math><mi>c</mi></math></span> such that every brace <span><math><mi>G</mi></math></span> has <span><math><mrow><mi>c</mi><mrow><mo>|</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow></math></span> thin edges. In this paper, we show that the subgraph induced by nonthin edges of <span><math><mi>G</mi></math></span>, two end vertices of which are of degree at least 4, is a forest. As an application, we show that every brace with <span><math><mrow><mi>k</mi><mrow><mo>|</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow></math></span> cubic vertices has at least <span><math><mrow><mrow><mo>(</mo><mn>2</mn><mo>−</mo><mn>5</mn><mi>k</mi><mo>)</mo></mrow><mrow><mo>|</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>/</mo><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math></span> thin edges, where <span><math><mrow><mn>0</mn><mo>&lt;</mo><mi>k</mi><mo>&lt;</mo><mn>0</mn><mo>.</mo><mn>4</mn></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"364 ","pages":"Pages 153-156"},"PeriodicalIF":1.0,"publicationDate":"2024-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143143036","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 algorithmic solutions for the Independent Roman Domination problem in graphs","authors":"Kaustav Paul,&nbsp;Ankit Sharma,&nbsp;Arti Pandey","doi":"10.1016/j.dam.2024.12.017","DOIUrl":"10.1016/j.dam.2024.12.017","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;, a function &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&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;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is said to be a &lt;em&gt;Roman Dominating function&lt;/em&gt; if for every &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; with &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;f&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;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, there exists a vertex &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;N&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;/mrow&gt;&lt;/math&gt;&lt;/span&gt; such that &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. A Roman Dominating function &lt;span&gt;&lt;math&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is said to be an &lt;em&gt;Independent Roman Dominating function&lt;/em&gt; (or IRDF), if &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;V&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;msub&gt;&lt;mrow&gt;&lt;mi&gt;V&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;/math&gt;&lt;/span&gt; forms an independent set, where &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&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;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;f&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;i&lt;/mi&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, for &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;i&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;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. The total weight of &lt;span&gt;&lt;math&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is equal to &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;f&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;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, and is denoted as &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. The &lt;em&gt;Independent Roman Domination Number&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;i&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 defined as min{&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is an IRDF of &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;}. For a given graph &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, the problem of computing &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;i&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 defined as the &lt;em&gt;Minimum Independent Roman Domination problem&lt;/em&gt;. The problem is already known to be NP-hard for bipartite graphs. In this paper, we further study the algorithmic complexity of the problem. In this paper, we propose a polynomial-time algorithm to solve the Minimum Independent Roman Domination problem for distance-heredita","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"364 ","pages":"Pages 143-152"},"PeriodicalIF":1.0,"publicationDate":"2024-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143143042","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":"Discrete isoperimetric method for bandwidth, pathwidth and treewidth of hypercubes","authors":"Lan Lin ,&nbsp;Yixun Lin","doi":"10.1016/j.dam.2024.12.001","DOIUrl":"10.1016/j.dam.2024.12.001","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;ω&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; denote the clique number of graph &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. The treewidth &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;tw&lt;/mo&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 &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is the minimum of &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;H&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;/mrow&gt;&lt;/math&gt;&lt;/span&gt; taken over all chordal supergraph &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&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;. The pathwidth &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;pw&lt;/mo&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 the bandwidth &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;bw&lt;/mo&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 defined in a similar way when the chordal graph &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is replaced by an interval graph or a proper interval graph respectively. It follows that &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;tw&lt;/mo&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;mo&gt;pw&lt;/mo&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;mo&gt;bw&lt;/mo&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;. A &lt;span&gt;&lt;math&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-dimensional hypercube &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; is the graph with vertex set of all &lt;span&gt;&lt;math&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-tuples &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;α&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;msub&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&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;α&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&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;mn&gt;1&lt;/mn&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; where two vertices are adjacent if they differ in exactly one coordinate. In order to determine the bandwidth &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mtext&gt;bw&lt;/mtext&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&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; of hypercubes, Harper (1966) proposed a powerful discrete isoperimetric method. Later, it was shown that &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;pw&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&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;bw&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&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;, but &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;tw&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&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; is unknown so far. In this paper, we review the discrete isoperimetric method for &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;bw&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&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;, &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;pw&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&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;, and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;tw&lt;/mo&gt;&lt;mr","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"363 ","pages":"Pages 201-214"},"PeriodicalIF":1.0,"publicationDate":"2024-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143098691","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 cyclic diagnosability of star graphs under the PMC and MM* models","authors":"Liu Mei ,&nbsp;Guo Chen ,&nbsp;Qiuming Liu ,&nbsp;Leng Ming","doi":"10.1016/j.dam.2024.12.005","DOIUrl":"10.1016/j.dam.2024.12.005","url":null,"abstract":"<div><div>Although traditional connectivity and diagnosability have become relatively mature in assessing the reliability of multiprocessor systems, they often inadequately capture specific nuanced characteristics of these systems. In order to achieve a more comprehensive evaluation of the diagnostic capability of interconnection networks, Zhang et al. introduced a novel metric termed cyclic diagnosability. Within a system <span><math><mi>G</mi></math></span>, the cyclic diagnosability of <span><math><mi>G</mi></math></span> represents the maximum cardinality of a faulty vertex set <span><math><mi>F</mi></math></span> that can be self-diagnosed, provided that <span><math><mrow><mi>G</mi><mo>−</mo><mi>F</mi></mrow></math></span> is disconnected and encompasses at least two cycles, with each cycle belonging to a different component. This paper presents an analysis of the structural properties of star graphs. Additional, we ascertain that the cyclic diagnosability of the <span><math><mi>n</mi></math></span>-dimensional star graph is <span><math><mrow><mn>7</mn><mi>n</mi><mo>−</mo><mn>20</mn></mrow></math></span>, under PMC model and MM* model for <span><math><mrow><mi>n</mi><mo>≥</mo><mn>13</mn></mrow></math></span>. The size is nearly seven times that of the traditional diagnosability of star graphs.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"364 ","pages":"Pages 60-73"},"PeriodicalIF":1.0,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143142512","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 maximum diversity of hypergraphs with fixed matching number","authors":"Peter Frankl ,&nbsp;Jian Wang","doi":"10.1016/j.dam.2024.12.012","DOIUrl":"10.1016/j.dam.2024.12.012","url":null,"abstract":"<div><div>Let <span><math><mrow><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow><mo>=</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></mrow></mrow></math></span> and let <span><math><mi>F</mi></math></span> be a family of <span><math><mi>k</mi></math></span>-subsets of <span><math><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></math></span>. The matching number of <span><math><mi>F</mi></math></span> is defined as the maximum number of pairwise disjoint members in <span><math><mi>F</mi></math></span>. The covering number of <span><math><mi>F</mi></math></span> is defined as the minimum size of <span><math><mrow><mi>T</mi><mo>⊂</mo><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mrow></math></span> such that each member of <span><math><mi>F</mi></math></span> intersects <span><math><mi>T</mi></math></span>. Define the <span><math><mi>t</mi></math></span>-diversity of <span><math><mi>F</mi></math></span> as the minimum size of <span><math><msup><mrow><mi>F</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> such that <span><math><mrow><mi>F</mi><mo>∖</mo><msup><mrow><mi>F</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math></span> has covering number <span><math><mi>t</mi></math></span>. Let <span><math><mi>F</mi></math></span> be a family of <span><math><mi>k</mi></math></span>-subsets of <span><math><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></math></span> with matching number <span><math><mi>s</mi></math></span>. In the present paper, we determine the maximum <span><math><mi>t</mi></math></span>-diversity of <span><math><mi>F</mi></math></span> for <span><math><mrow><mn>1</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>2</mn><mi>s</mi><mo>−</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>n</mi><mo>≥</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>k</mi><mo>,</mo><mi>s</mi><mo>)</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"364 ","pages":"Pages 120-135"},"PeriodicalIF":1.0,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143142511","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}