Discrete Mathematics最新文献

{"title":"Six-flows of signed graphs with frustration index three","authors":"You Lu ,&nbsp;Rong Luo ,&nbsp;Cun-Quan Zhang","doi":"10.1016/j.disc.2024.114325","DOIUrl":"10.1016/j.disc.2024.114325","url":null,"abstract":"<div><div>Bouchet's 6-flow conjecture states that every flow-admissible signed graph admits a nowhere-zero 6-flow. Seymour's 6-flow theorem states that the conjecture holds for balanced signed graphs. Rollová et al. show that every flow-admissible signed graph with frustration index two admits a nowhere-zero 7-flow, where the frustration index of a signed graph is the smallest number of edges whose deletion leaves a balanced signed graph. Wang et al. improve the result to 6-flows. In this paper, we further extend these results, and confirm Bouchet's 6-flow conjecture for signed graphs with frustration index three. There are infinitely many signed graphs with frustration index three admitting a nowhere-zero 6-flow but no 5-flow. From the point of view of flow theory, signed graphs with frustration index two are very similar to those of ordinary graphs. However, there are significant differences between ordinary graphs and signed graphs with frustration index three.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114325"},"PeriodicalIF":0.7,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142655718","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":"Partial packing coloring and quasi-packing coloring of the triangular grid","authors":"Hubert Grochowski,&nbsp;Konstanty Junosza-Szaniawski","doi":"10.1016/j.disc.2024.114308","DOIUrl":"10.1016/j.disc.2024.114308","url":null,"abstract":"<div><div>The concept of packing coloring in graph theory is motivated by the challenge of frequency assignment in radio networks. This approach entails assigning positive integers to vertices, with the requirement that for any given label (color) <em>i</em>, the distance between any two vertices sharing this label must exceed <em>i</em>. Recently, after over 20 years of intensive research, the minimal number of colors needed for packing coloring of an infinite square grid has been established to be 15. Moreover, it is known that a hexagonal grid requires a minimum of 7 colors for packing coloring, and a triangular grid is not colorable with any finite number of colors in a packing way.</div><div>Therefore, two questions come to mind: What fraction of a triangular grid can be colored in a packing model, and how much do we need to weaken the condition of packing coloring to enable coloring a triangular grid with a finite number of colors?</div><div>With the partial help of the Mixed Integer Linear Programming (MILP) solver, we have proven that it is possible to color at least 72.8% but no more than 82.2% of a triangular grid in a packing way.</div><div>Additionally, we have investigated the relaxation of packing coloring, called quasi-packing coloring, which is a special case of <em>S</em>-packing coloring. We have established that the <em>S</em>-packing chromatic number for the triangular grid, where <span><math><mi>S</mi><mo>=</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>)</mo></math></span>, is between 11 and 33. Furthermore, we have proven that the aforementioned sequence <em>S</em> is the best possible in some sense.</div><div>We have also considered the partial packing and quasi-packing coloring of an infinite hypercube and present several open problems for other classes of graphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114308"},"PeriodicalIF":0.7,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142655714","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 Mneimneh sums and their application to multiple polylogarithms","authors":"Marian Genčev","doi":"10.1016/j.disc.2024.114318","DOIUrl":"10.1016/j.disc.2024.114318","url":null,"abstract":"<div><div>The purpose of this paper is the study of the binomial sum<span><span><span><math><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>⋅</mo><msubsup><mrow><mi>H</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></msubsup><mo>(</mo><mi>a</mi><mo>)</mo><mo>⋅</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>⋅</mo><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>p</mi><mo>)</mo></mrow><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup><mo>,</mo></math></span></span></span> where <span><math><msubsup><mrow><mi>H</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></msubsup><mo>(</mo><mi>a</mi><mo>)</mo><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>k</mi></mrow></msubsup><msup><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msup><mo>/</mo><msup><mrow><mi>i</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span> denotes the parameterized analogue of the <em>k</em>-th harmonic number of order <em>s</em>. For <span><math><mi>a</mi><mo>=</mo><mi>s</mi><mo>=</mo><mn>1</mn></math></span>, these binomial sums were investigated by Mneimneh, who gave a probabilistic interpretation related to hiring problems. We present a generalization of Mneimneh's summation formula and establish several new identities and a connection of these sums with specific multiple polylogarithms, called unit Euler sums, based upon the Toeplitz limit theorem.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114318"},"PeriodicalIF":0.7,"publicationDate":"2024-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142655324","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 combinatorial and hypergeometric approaches toward second-order difference equations","authors":"John M. Campbell","doi":"10.1016/j.disc.2024.114316","DOIUrl":"10.1016/j.disc.2024.114316","url":null,"abstract":"<div><div>Laohakosol et al. recently introduced enumerative techniques based on second-order difference equations to prove a number of conjectured evaluations for polynomial continued fractions generated by the Ramanujan Machine. Each of the discrete difference equations required according to the combinatorial approach employed by Laohakosol et al. can be solved in an <em>explicit</em> way according to an alternative and hypergeometric-based approach that we apply to prove further conjectures produced by the Ramanujan Machine. An advantage of our hypergeometric approach, compared to the methods of Laohakosol et al. and compared to solving for ODEs satisfied by formal power series corresponding to the Euler–Wallis recursions, is given by the explicit evaluations for the nonlinear difference equations that we obtain.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114316"},"PeriodicalIF":0.7,"publicationDate":"2024-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142655717","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":"Packing directed cycles of specified odd length into digraphs and alternating cycles into bipartite graphs","authors":"Shuya Chiba ,&nbsp;Koshin Yoshida","doi":"10.1016/j.disc.2024.114306","DOIUrl":"10.1016/j.disc.2024.114306","url":null,"abstract":"<div><div>In this paper, we prove the following result. For given integers <span><math><mi>k</mi><mo>≥</mo><mn>1</mn><mo>,</mo><mspace></mspace><mi>t</mi><mo>≥</mo><mn>0</mn></math></span> with <span><math><mi>k</mi><mo>≥</mo><mi>t</mi></math></span> and an odd integer <span><math><mi>ℓ</mi><mo>≥</mo><mn>3</mn></math></span>, there exists an integer <span><math><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>k</mi><mo>,</mo><mi>t</mi><mo>,</mo><mi>ℓ</mi><mo>)</mo></math></span> satisfying the following: If <em>D</em> is a digraph of order <span><math><mi>n</mi><mo>≥</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>, and if <span><math><msubsup><mrow><mi>d</mi></mrow><mrow><mi>D</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>(</mo><mi>u</mi><mo>)</mo><mo>+</mo><msubsup><mrow><mi>d</mi></mrow><mrow><mi>D</mi></mrow><mrow><mo>−</mo></mrow></msubsup><mo>(</mo><mi>v</mi><mo>)</mo><mo>≥</mo><mi>n</mi><mo>+</mo><mi>t</mi></math></span> for every two distinct vertices <em>u</em> and <em>v</em> with <span><math><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo><mo>∉</mo><mi>A</mi><mo>(</mo><mi>D</mi><mo>)</mo></math></span>, then <em>D</em> contains <em>k</em> vertex-disjoint directed cycles of length <em>ℓ</em> or <span><math><mi>ℓ</mi><mo>+</mo><mn>1</mn></math></span> such that at least <em>t</em> of them are of length <em>ℓ</em>. This is a common extension of the results obtained by Brandt et al. (1997) and, Chiba and Yamashita (2018). We also discuss the relation between our result and problems on packing alternating cycles into bipartite graphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114306"},"PeriodicalIF":0.7,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142593040","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":"Refinements of degree conditions for the existence of a spanning tree without small degree stems","authors":"Michitaka Furuya ,&nbsp;Akira Saito ,&nbsp;Shoichi Tsuchiya","doi":"10.1016/j.disc.2024.114307","DOIUrl":"10.1016/j.disc.2024.114307","url":null,"abstract":"<div><div>A spanning tree of a graph without vertices of degree 2 is called a <em>homeomorphically irreducible spanning tree</em> (or a <em>HIST</em>) of the graph. Albertson et al. (1990) <span><span>[1]</span></span> gave a minimum degree condition for the existence of a HIST, and recently, Ito and Tsuchiya (2022) <span><span>[11]</span></span> found a sharp degree-sum condition for the existence of a HIST. In this paper, we refine these results, and extend the first one to a spanning tree in which no vertex other than the endvertices has small degree.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114307"},"PeriodicalIF":0.7,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142587366","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":"Every 3-connected {K1,3,Γ3}-free graph is Hamilton-connected","authors":"Adam Kabela,&nbsp;Zdeněk Ryjáček,&nbsp;Mária Skyvová,&nbsp;Petr Vrána","doi":"10.1016/j.disc.2024.114305","DOIUrl":"10.1016/j.disc.2024.114305","url":null,"abstract":"<div><div>We show that every 3-connected <span><math><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Γ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>}</mo></math></span>-free graph is Hamilton-connected, where <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> is the graph obtained by joining two vertex-disjoint triangles with a path of length 3. This resolves one of the two last open cases in the characterization of pairs of connected forbidden subgraphs implying Hamilton-connectedness. The proof is based on a new closure technique, developed in a previous paper, and on a structural analysis of small subgraphs, cycles and paths in line graphs of multigraphs. The most technical steps of the analysis are computer-assisted.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114305"},"PeriodicalIF":0.7,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142560889","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Completely regular codes with covering radius 1 and the second eigenvalue in 3-dimensional Hamming graphs","authors":"Ivan Mogilnykh,&nbsp;Anna Taranenko,&nbsp;Konstantin Vorob'ev","doi":"10.1016/j.disc.2024.114296","DOIUrl":"10.1016/j.disc.2024.114296","url":null,"abstract":"<div><div>We obtain a classification of completely regular codes with covering radius 1 and the second eigenvalue in the Hamming graphs <span><math><mi>H</mi><mo>(</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> up to <em>q</em> and intersection array. Due to the works of Meyerowitz, Mogilnykh, and Valyuzenich, our result completes the classifications of completely regular codes with covering radius 1 and the second eigenvalue in the Hamming graphs <span><math><mi>H</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span> for any <em>n</em> and completely regular codes with covering radius 1 in <span><math><mi>H</mi><mo>(</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114296"},"PeriodicalIF":0.7,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142560885","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 finding the largest minimum distance of locally recoverable codes: A graph theory approach","authors":"Majid Khabbazian ,&nbsp;Muriel Médard","doi":"10.1016/j.disc.2024.114298","DOIUrl":"10.1016/j.disc.2024.114298","url":null,"abstract":"<div><div>The <span><math><mo>[</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>r</mi><mo>]</mo></math></span>-Locally recoverable codes (LRC) studied in this work are a well-studied family of <span><math><mo>[</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>]</mo></math></span> linear codes for which the value of each symbol can be recovered by a linear combination of at most <em>r</em> other symbols. In this paper, we study the <em>LMD</em> problem, which is to find the largest possible minimum distance of <span><math><mo>[</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>r</mi><mo>]</mo></math></span>-LRCs, denoted by <span><math><mi>D</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span>. LMD can be approximated within an additive term of one—it is known that <span><math><mi>D</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span> is equal to either <span><math><msup><mrow><mi>d</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> or <span><math><msup><mrow><mi>d</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>−</mo><mn>1</mn></math></span>, where <span><math><msup><mrow><mi>d</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>=</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo>−</mo><mrow><mo>⌈</mo><mfrac><mrow><mi>k</mi></mrow><mrow><mi>r</mi></mrow></mfrac><mo>⌉</mo></mrow><mo>+</mo><mn>2</mn></math></span>. Moreover, for a range of parameters, it is known precisely whether the distance <span><math><mi>D</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span> is <span><math><msup><mrow><mi>d</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> or <span><math><msup><mrow><mi>d</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>−</mo><mn>1</mn></math></span>. However, the problem is still open despite a significant effort. In this work, we convert LMD to an equivalent simply-stated problem in graph theory. Using this conversion, we show that an instance of LMD is at least as hard as computing the size of a maximal graph of high girth, a hard problem in extremal graph theory. This is an evidence that LMD—although can be approximated within an additive term of one—is hard to solve in general. As a positive result, thanks to the conversion and the exiting results in extremal graph theory, we solve LMD for a range of code parameters that has not been solved before.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114298"},"PeriodicalIF":0.7,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142560887","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":"Extremal bounds for pattern avoidance in multidimensional 0-1 matrices","authors":"Jesse Geneson ,&nbsp;Shen-Fu Tsai","doi":"10.1016/j.disc.2024.114303","DOIUrl":"10.1016/j.disc.2024.114303","url":null,"abstract":"&lt;div&gt;&lt;div&gt;A 0-1 matrix &lt;em&gt;M&lt;/em&gt; contains another 0-1 matrix &lt;em&gt;P&lt;/em&gt; if some submatrix of &lt;em&gt;M&lt;/em&gt; can be turned into &lt;em&gt;P&lt;/em&gt; by changing any number of 1-entries to 0-entries. The 0-1 matrix &lt;em&gt;M&lt;/em&gt; is &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-saturated where &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is a family of 0-1 matrices if &lt;em&gt;M&lt;/em&gt; avoids every element of &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and changing any 0-entry of &lt;em&gt;M&lt;/em&gt; to a 1-entry introduces a copy of some element of &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. The extremal function &lt;span&gt;&lt;math&gt;&lt;mi&gt;ex&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and saturation function &lt;span&gt;&lt;math&gt;&lt;mi&gt;sat&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; are the maximum and minimum possible number of 1-entries in an &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-saturated 0-1 matrix, respectively, and the semisaturation function &lt;span&gt;&lt;math&gt;&lt;mi&gt;ssat&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is the minimum possible number of 1-entries in an &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-semisaturated 0-1 matrix &lt;em&gt;M&lt;/em&gt;, i.e., changing any 0-entry in &lt;em&gt;M&lt;/em&gt; to a 1-entry introduces a new copy of some element of &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;.&lt;/div&gt;&lt;div&gt;We study these functions of multidimensional 0-1 matrices. In particular, we give upper bounds on parameters of minimally non-&lt;span&gt;&lt;math&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; &lt;em&gt;d&lt;/em&gt;-dimensional 0-1 matrices, generalized from minimally nonlinear 0-1 matrices in two dimensions, and we show the existence of infinitely many minimally non-&lt;span&gt;&lt;math&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; &lt;em&gt;d&lt;/em&gt;-dimensional 0-1 matrices with all dimensions of length greater than 1. For any positive integers &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and integer &lt;span&gt;&lt;math&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, we construct a family of &lt;em&gt;d&lt;/em&gt;-dimensional 0-1 matrices with both extremal function and saturation function exactly &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; for sufficiently large &lt;em&gt;n&lt;/em&gt;. We show that no family of &lt;em&gt;d&lt;/em&gt;-dimensional 0-1 matrices has saturation function strictly between &lt;span&gt;&lt;math&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;Θ&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and we construct a family of &lt;em&gt;d&lt;/em&gt;-dimensional 0-1 matrices with bounded saturation function and extremal function &lt;span&gt;&lt;math&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;ϵ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/spa","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114303"},"PeriodicalIF":0.7,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142555007","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}