{"title":"Oriented colouring graphs of bounded degree and degeneracy","authors":"A. Clow, L. Stacho","doi":"10.1016/j.disc.2025.114746","DOIUrl":"10.1016/j.disc.2025.114746","url":null,"abstract":"<div><div>This paper considers upper bounds on the oriented chromatic number <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>o</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, of an oriented graph <em>G</em> in terms of its 2-dipath chromatic number <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, degeneracy <span><math><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, and maximum degree <span><math><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. In particular, we show that for all graphs <em>G</em> with <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>k</mi></math></span> where <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> and <span><math><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>t</mi></math></span> where <span><math><mi>t</mi><mo>≥</mo><msub><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msub><mo></mo><mo>(</mo><mi>k</mi><mo>)</mo></math></span>, <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>o</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mn>33</mn><mo>/</mo><mn>10</mn><mo>(</mo><mi>k</mi><msup><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mn>2</mn></mrow><mrow><mi>t</mi></mrow></msup><mo>)</mo></math></span>. This improves an upper bound of MacGillivray, Raspaud, and Swartz of the form <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>o</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msup><mo>−</mo><mn>1</mn></math></span> to a polynomial upper bound for many classes of graphs, in particular, those with bounded degeneracy. Additionally, we asymptotically improve bounds for the oriented chromatic number in terms of maximum degree and degeneracy. For instance, we show that <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>o</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mo>(</mo><mn>2</mn><mi>ln</mi><mo></mo><mn>2</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><msup><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mn>2</mn></mrow><mrow><mi>Δ</mi></mrow></msup></math></span> for all graphs, and <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>o</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mo>(</mo><mn>2</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><mi>Δ</mi><mi>d</mi><msup><mrow><mn>2</mn></mrow><mrow><mi>d</mi></mrow></msup></math></span> for graphs where degeneracy grows sublinearly in maximum degree. Here the asymptotics are in Δ. The former improves the asymptotics of a results by Kostochka, Sopena, and Zhu <span><span>[9]</span></span>, while the latter improves the asymptotics of a result by Aravind and Subramanian <span><span>[1]</span></span>. Both improvements are by a constant factor.</","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114746"},"PeriodicalIF":0.7,"publicationDate":"2025-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144885662","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}
Dan Zeng , Yunge Xu , Lisha Li , Xianping Liu , Xiangyong Zeng
{"title":"The cycle structure of a class of permutation binomials","authors":"Dan Zeng , Yunge Xu , Lisha Li , Xianping Liu , Xiangyong Zeng","doi":"10.1016/j.disc.2025.114740","DOIUrl":"10.1016/j.disc.2025.114740","url":null,"abstract":"<div><div>In this paper, we study the cycle structure of permutation binomials of the form <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>θ</mi><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>+</mo><mn>1</mn></mrow></msup></math></span> with 4-uniform DDT and 4-uniform BCT over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msup></mrow></msub></math></span>. Note that the <em>k</em>-th composition of <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> can be rewritten as the square root of a product of two polynomials <span><math><msub><mrow><mi>α</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mrow><mo>(</mo><msup><mrow><mi>f</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow></msup></math></span> and <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mrow><mo>(</mo><msup><mrow><mi>f</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></mrow><mrow><mn>1</mn><mo>−</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></mrow></msup></math></span> for any <span><math><mi>x</mi><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msup></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span>, that is, <span><math><msup><mrow><mi>f</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mrow><mo>(</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></math></span>. By further studying the periods of <span><math><msub><mrow><mi>α</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span>, we get the cycle structure of <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msup></mrow></msub></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114740"},"PeriodicalIF":0.7,"publicationDate":"2025-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144885661","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":"Splitting fields of Schreier digraphs and t-Cayley hypergraphs","authors":"Yongjiang Wu , Weijun Liu , Lihua Feng , Xiaoqian Zhang","doi":"10.1016/j.disc.2025.114742","DOIUrl":"10.1016/j.disc.2025.114742","url":null,"abstract":"<div><div>In this paper, we determine the splitting fields and algebraic degrees of normal Schreier digraphs and certain <em>t</em>-Cayley hypergraphs. Additionally, we derive a decomposition formula for the spectra and distance spectra of <em>t</em>-Cayley hypergraphs over arbitrary finite groups, respectively. Furthermore, we determine the distance splitting fields and distance algebraic degrees of normal <em>t</em>-Cayley hypergraphs. These results generalize several previous work by Lu and Mönius on Cayley graphs and Sripaisan and Meemark on <em>t</em>-Cayley hypergraphs regarding splitting fields and algebraic degrees.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114742"},"PeriodicalIF":0.7,"publicationDate":"2025-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144885728","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":"Four families of squares of BCH codes and their complements","authors":"Bin Zheng, Shixin Zhu","doi":"10.1016/j.disc.2025.114736","DOIUrl":"10.1016/j.disc.2025.114736","url":null,"abstract":"<div><div>The Schur square code <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> of a linear code <span><math><mi>C</mi></math></span> is the linear code spanned by all component-wise products of two elements of <span><math><mi>C</mi></math></span>. Schur square codes play an essential role in solving cryptographic problems, especially secure multiparty computation and <em>t</em>-strongly multiplicative secret sharing schemes. In this paper, we are focused on the squares of BCH codes and their complements, which are a subclass of cyclic codes and have good error-correcting capability and wide applications. We present necessary and sufficient conditions to guarantee that the squares of narrow-sense and non-narrow-sense BCH codes and their complements are respectively not equal to <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>. In addition, we investigate the squares of the BCH codes with special designed distances and determine their explicit dimensions and lower bounds on minimum distances.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114736"},"PeriodicalIF":0.7,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144880001","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"An upper bound on the number of relevant variables for Boolean functions on the Hamming graph","authors":"Alexandr Valyuzhenich","doi":"10.1016/j.disc.2025.114745","DOIUrl":"10.1016/j.disc.2025.114745","url":null,"abstract":"<div><div>The spectrum of a complex-valued function <em>f</em> on <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> is the set <span><math><mo>{</mo><mo>|</mo><mi>u</mi><mo>|</mo><mo>:</mo><mi>u</mi><mo>∈</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msubsup><mspace></mspace><mrow><mi>and</mi></mrow><mspace></mspace><mover><mrow><mi>f</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>(</mo><mi>u</mi><mo>)</mo><mo>≠</mo><mn>0</mn><mo>}</mo></math></span>, where <span><math><mo>|</mo><mi>u</mi><mo>|</mo></math></span> is the Hamming weight of <em>u</em> and <span><math><mover><mrow><mi>f</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span> is the Fourier transform of <em>f</em>. Let <span><math><mn>1</mn><mo>≤</mo><msup><mrow><mi>d</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>≤</mo><mi>d</mi><mo>≤</mo><mi>n</mi></math></span>. In this work, we study Boolean functions on <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>, <span><math><mi>q</mi><mo>≥</mo><mn>3</mn></math></span>, whose spectrum is a subset of <span><math><mo>{</mo><mn>0</mn><mo>}</mo><mo>∪</mo><mo>{</mo><msup><mrow><mi>d</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>,</mo><mo>…</mo><mo>,</mo><mi>d</mi><mo>}</mo></math></span>. We prove that such functions have at most <span><math><mfrac><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⋅</mo><mfrac><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>d</mi><mo>+</mo><msup><mrow><mi>d</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msup></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><msup><mrow><mi>d</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msup><msup><mrow><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><msup><mrow><mi>d</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msup></mrow></mfrac></math></span> relevant variables for <span><math><msup><mrow><mi>d</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>+</mo><mi>d</mi><mo>≤</mo><mi>n</mi><mo>+</mo><mn>1</mn></math></span>. In particular, we prove that any Boolean function of degree <em>d</em> on <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>, <span><math><mi>q</mi><mo>≥</mo><mn>3</mn></math></span>, has at most <span><math><mfrac><mrow><mi>d</mi><msup><mrow><mi>q</mi></mrow><mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><mrow><mn>4</mn><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mfrac></math></span> relevant variables. We also show that any equitable 2-partition of the Hamming graph <span><math><mi>H</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, <span><math><mi>q</mi><mo>≥</mo><mn>3</mn></math></span>, associated with the eigenvalue <span><math><mi>n</mi><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>−</mo><mi>q</mi><mi>d</mi></math></span> has at most <span><math><mfrac><mr","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114745"},"PeriodicalIF":0.7,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144880005","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}
Hengzhe Li , Zhiwei Ding , Jianbing Liu , Hong-Jian Lai
{"title":"Diameter two orientability of mixed graphs","authors":"Hengzhe Li , Zhiwei Ding , Jianbing Liu , Hong-Jian Lai","doi":"10.1016/j.disc.2025.114732","DOIUrl":"10.1016/j.disc.2025.114732","url":null,"abstract":"<div><div>In 1967, Katona and Szemerédi showed that no undirected graph with <em>n</em> vertices and fewer than <span><math><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><msub><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msub><mo></mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> edges admits an orientation of diameter two. In 1978, Chvátal and Thomassen revealed the complexity of determining whether an undirected graph can be oriented to achieve a diameter of two, proving it to be NP-complete. This breakthrough has sparked ongoing interest in identifying sufficient conditions for graphs to be oriented with the smallest possible diameter of two—critical for optimizing communication and network flow in larger structures. In 1985, Chung, Garey, and Tarjan extended the work of Chvátal and Thomassen. In 2019, Czabarka, Dankelmann, and Székely significantly advanced this field by establishing that the minimum degree threshold for achieving such an orientation in undirected graphs of order <em>n</em> is <span><math><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>Θ</mi><mo>(</mo><mi>ln</mi><mo></mo><mi>n</mi><mo>)</mo></math></span>. In this paper, we extend this foundational result by determining the minimum degree threshold necessary for realizing an orientation with diameter two in mixed graphs, which contain both undirected and directed edges. Mixed graphs offer a versatile framework, representing an intermediate stage in the orientation process, making our findings a substantial generalization of previous results.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 1","pages":"Article 114732"},"PeriodicalIF":0.7,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144878766","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 second-order zero differential uniformity of the swapped inverse functions over finite fields","authors":"Jaeseong Jeong , Namhun Koo , Soonhak Kwon","doi":"10.1016/j.disc.2025.114738","DOIUrl":"10.1016/j.disc.2025.114738","url":null,"abstract":"<div><div>The Feistel Boomerang Connectivity Table (FBCT) was proposed as the Feistel counterpart of the Boomerang Connectivity Table. The entries of the FBCT are actually related to the second-order zero differential spectrum. Recently, several results on the second-order zero differential uniformity of some functions were introduced. However, almost all of them were focused on power functions, and there are only a few results on non-power functions. In this paper, we investigate the second-order zero differential uniformity of the swapped inverse functions, which are functions obtained from swapping two points in the inverse function. We also present the second-order zero differential spectrum of the swapped inverse functions for certain cases. In particular, this paper is the first result to characterize classes of non-power functions with the second-order zero differential uniformity equal to 4, in even characteristic.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 1","pages":"Article 114738"},"PeriodicalIF":0.7,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144878158","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 existence and non-existence of spherical m-stiff configurations","authors":"Eiichi Bannai , Hirotake Kurihara , Hiroshi Nozaki","doi":"10.1016/j.disc.2025.114731","DOIUrl":"10.1016/j.disc.2025.114731","url":null,"abstract":"<div><div>This paper investigates the existence of <em>m</em>-stiff configurations in the unit sphere <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>, which are spherical <span><math><mo>(</mo><mn>2</mn><mi>m</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-designs that lie on <em>m</em> parallel hyperplanes. We establish two non-existence results: (1) for each fixed integer <span><math><mi>m</mi><mo>></mo><mn>5</mn></math></span>, there exists no <em>m</em>-stiff configuration in <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> for sufficiently large <em>d</em>; (2) for each fixed integer <span><math><mi>d</mi><mo>></mo><mn>10</mn></math></span>, there exists no <em>m</em>-stiff configuration in <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> for sufficiently large <em>m</em>. Furthermore, we provide a complete classification of the dimensions where <em>m</em>-stiff configurations exist for <span><math><mi>m</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn></math></span>. We also determine the non-existence (and the existence) of <em>m</em>-stiff configurations in <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> for small <em>d</em> (<span><math><mn>3</mn><mo>≤</mo><mi>d</mi><mo>≤</mo><mn>120</mn></math></span>) with arbitrary <em>m</em>, and also for small <em>m</em> (<span><math><mn>6</mn><mo>≤</mo><mi>m</mi><mo>≤</mo><mn>10</mn></math></span>) with arbitrary <em>d</em>. Finally, we conjecture that there is no <em>m</em>-stiff configuration in <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> for <span><math><mo>(</mo><mi>d</mi><mo>,</mo><mi>m</mi><mo>)</mo></math></span> with <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span> and <span><math><mi>m</mi><mo>≥</mo><mn>6</mn></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 1","pages":"Article 114731"},"PeriodicalIF":0.7,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144878159","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 order ideals in orbit codes of finite Abelian groups: Results and impacts","authors":"Sihem Mesnager , Rameez Raja","doi":"10.1016/j.disc.2025.114727","DOIUrl":"10.1016/j.disc.2025.114727","url":null,"abstract":"<div><div><em>Automorphism orbit codes</em> associated with finite Abelian groups represent an extended form of the widely recognized <em>homomorphism codes</em>. The article explores the characteristics and attributes of <em>order ideals</em> within an <em>automorphism orbit code</em> <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>λ</mi></mrow></msub></math></span>, where <em>λ</em> is a partition of a positive integer. This connection between <em>order ideal</em> points and binary codewords is significant because it highlights the adaptability of poset theory in addressing real-world challenges in information theory, coding theory, and cryptography. The study aims to uncover the intrinsic properties that shape the role and impact of these ideals within the poset structure. The article provides an explicit formula to compute the cardinality of the set of ideals in <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>λ</mi></mrow></msub></math></span> and shows that the cardinality of a specific code called the <em>signature code</em> in <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>λ</mi></mrow></msub></math></span> is a power of a prime <em>p</em>. Additionally, it presents codes with a cardinality expressible as a polynomial in <em>p</em> with integer coefficients. Finally, the article concludes by discussing characteristic subgroups of Abelian <em>p</em>-groups and proving that codes within <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>λ</mi></mrow></msub></math></span> are dense.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 1","pages":"Article 114727"},"PeriodicalIF":0.7,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144878932","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":"Maximum group divisible packings with block size four","authors":"Lidong Wang , Qi Feng , Pinpin Zhou , Aixin Chen , Zihong Tian","doi":"10.1016/j.disc.2025.114733","DOIUrl":"10.1016/j.disc.2025.114733","url":null,"abstract":"<div><div>In this paper, we consider the constructions for group divisible packings of type <span><math><msup><mrow><mi>g</mi></mrow><mrow><mi>u</mi></mrow></msup></math></span> with block size four (4-GDPs for short). A maximum 4-GDP of type <span><math><msup><mrow><mi>g</mi></mrow><mrow><mi>u</mi></mrow></msup></math></span> contains the largest possible number of blocks. Based on recursive constructions, we establish a framework to construct maximum 4-GDPs of type <span><math><msup><mrow><mi>g</mi></mrow><mrow><mi>u</mi></mrow></msup></math></span>. In the process, direct constructions on several key auxiliary designs are displayed by choosing appropriate automorphism groups. Eventually, the sizes of maximum 4-GDPs of type <span><math><msup><mrow><mi>g</mi></mrow><mrow><mi>u</mi></mrow></msup></math></span> are determined for all positive integers <em>g</em> and <em>u</em>, only leaving a small fraction of possible exceptions unresolved.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114733"},"PeriodicalIF":0.7,"publicationDate":"2025-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144863574","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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