Discrete Mathematics最新文献

{"title":"The c-boomerang uniformity and c-boomerang spectrum of two classes of permutation polynomials over the finite field F2n","authors":"Guanghui Li ,&nbsp;Xiwang Cao","doi":"10.1016/j.disc.2025.114543","DOIUrl":"10.1016/j.disc.2025.114543","url":null,"abstract":"<div><div>The boomerang attack developed by Wagner is a cryptanalysis technique against block ciphers. A new theoretical tool, the Boomerang Connectivity Table (BCT) and the corresponding boomerang uniformity were introduced to evaluate the resistance of a block cipher against the boomerang attack. Using a multiplier differential, Stănică (2021) <span><span>[28]</span></span> extended the notion of boomerang uniformity to <em>c</em>-boomerang uniformity. In this paper, we focus on two classes of permutation polynomials over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span>. For one of these, we show that the <em>c</em>-boomerang uniformity of this function is equal to 1. For the second type of function, we first consider the <em>c</em>-BCT entries. We then explicitly determine the <em>c</em>-boomerang spectrum of this function by means of characters and some techniques in solving equations over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 9","pages":"Article 114543"},"PeriodicalIF":0.7,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143825675","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 unified approach to the spectral radius, connectivity and edge-connectivity of graphs","authors":"Yu Wang ,&nbsp;Dan Li ,&nbsp;Huiqiu Lin","doi":"10.1016/j.disc.2025.114544","DOIUrl":"10.1016/j.disc.2025.114544","url":null,"abstract":"<div><div>For two integers <span><math><mi>r</mi><mo>≥</mo><mn>2</mn></math></span> and <span><math><mi>h</mi><mo>≥</mo><mn>0</mn></math></span>, the <em>h-extra r-component connectivity</em> <span><math><msubsup><mrow><mi>κ</mi></mrow><mrow><mi>r</mi></mrow><mrow><mi>h</mi></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of a graph <em>G</em> is defined as the minimum size of a subset <em>S</em> of vertices whose removal disconnects <em>G</em>, such that there are at least <em>r</em> connected components in <span><math><mi>G</mi><mo>−</mo><mi>S</mi></math></span> and each component has at least <span><math><mi>h</mi><mo>+</mo><mn>1</mn></math></span> vertices. Denote by <span><math><msubsup><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>δ</mi></mrow><mrow><msubsup><mrow><mi>κ</mi></mrow><mrow><mi>r</mi></mrow><mrow><mi>h</mi></mrow></msubsup></mrow></msubsup></math></span> the set of <em>n</em>-vertex graphs with <em>h</em>-extra <em>r</em>-component connectivity <span><math><msubsup><mrow><mi>κ</mi></mrow><mrow><mi>r</mi></mrow><mrow><mi>h</mi></mrow></msubsup></math></span> and minimum degree <em>δ</em>. The following problem concerning spectral radius was proposed by Brualdi and Solheid (1986) <span><span>[2]</span></span>: Given a set of graphs <span><math><mi>S</mi></math></span>, find an upper bound for the spectral radius of graphs in <span><math><mi>S</mi></math></span> and characterize the graphs in which the maximum spectral radius is attained. We study this question for <span><math><mi>S</mi><mo>=</mo><msubsup><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>δ</mi></mrow><mrow><msubsup><mrow><mi>κ</mi></mrow><mrow><mi>r</mi></mrow><mrow><mi>h</mi></mrow></msubsup></mrow></msubsup></math></span> where <span><math><mi>r</mi><mo>≥</mo><mn>2</mn></math></span> and <span><math><mi>h</mi><mo>≥</mo><mn>0</mn></math></span>. Fan, Gu and Lin (2024) <span><span>[7]</span></span> answered the question for <span><math><mi>r</mi><mo>≥</mo><mn>2</mn></math></span> and <span><math><mi>h</mi><mo>=</mo><mn>0</mn></math></span>. In this paper, we solve this problem completely for <span><math><mi>r</mi><mo>≥</mo><mn>2</mn></math></span> and <span><math><mi>h</mi><mo>≥</mo><mn>1</mn></math></span>. Moreover, we also investigate analogous problems for the edge version. This implies some previous results in connectivity and edge-connectivity.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 9","pages":"Article 114544"},"PeriodicalIF":0.7,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143825674","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 family of self-orthogonal divisible codes with locality 2","authors":"Ziling Heng ,&nbsp;Mengjie Yang ,&nbsp;Yang Ming","doi":"10.1016/j.disc.2025.114529","DOIUrl":"10.1016/j.disc.2025.114529","url":null,"abstract":"<div><div>Linear codes are widely studied due to their applications in communication, cryptography, quantum codes, distributed storage and many other fields. In this paper, we use the trace and norm functions over finite fields to construct a family of linear codes. The weight distributions of the codes are determined in three cases via Gaussian sums. The codes are shown to be self-orthogonal divisible codes with only three, four or five nonzero weights in these cases. In particular, we prove that this family of linear codes has locality 2. Several optimal or almost optimal linear codes and locally recoverable codes are derived. In particular, an infinite family of distance-optimal binary linear codes with respect to the sphere-packing bound is obtained. The self-orthogonal codes derived in this paper can be used to construct lattices and have nice application in distributed storage.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 10","pages":"Article 114529"},"PeriodicalIF":0.7,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143826399","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":"Bounds in radial Moore graphs of diameter 3","authors":"Jesús M. Ceresuela,&nbsp;Nacho López","doi":"10.1016/j.disc.2025.114533","DOIUrl":"10.1016/j.disc.2025.114533","url":null,"abstract":"<div><div>Radial Moore graphs preserve the order and the regularity of Moore graphs and allow some vertices to have more eccentricity than they should have in a Moore graph. One way to classify their resemblance with a Moore graph is the status measure. The status of a graph is defined as the sum of the distances of all pairs of ordered vertices and equals twice the Wiener index. Vertices with minimum eccentricity are called central vertices. In this paper we study upper bounds for both the maximum number of central vertices and the status of radial Moore graphs. Finally, we present a family of radial Moore graphs of diameter 3 that is conjectured to have maximum status.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 9","pages":"Article 114533"},"PeriodicalIF":0.7,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143817466","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":"Minimum saturated graphs for unions of cliques","authors":"Wen-Han Zhu,&nbsp;Rong-Xia Hao,&nbsp;Zhen He","doi":"10.1016/j.disc.2025.114530","DOIUrl":"10.1016/j.disc.2025.114530","url":null,"abstract":"<div><div>Let <em>H</em> be a fixed graph. A graph <em>G</em> is called <em>H-saturated</em> if <em>G</em> does not contain a subgraph isomorphic to <em>H</em>, but the addition of any missing edge to <em>G</em> results in a copy of <em>H</em> in <em>G</em>. The <em>saturation number</em> of <em>H</em>, denoted <span><math><mi>s</mi><mi>a</mi><mi>t</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span>, is the minimum number of edges among all <em>H</em>-saturated graphs of order <em>n</em>, and <span><math><mi>S</mi><mi>a</mi><mi>t</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> denotes the family of <em>H</em>-saturated graphs with <span><math><mi>s</mi><mi>a</mi><mi>t</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> edges and <em>n</em> vertices. In this paper, we resolve a conjecture of Chen and Yuan (2024) <span><span>[4]</span></span> by determining <span><math><mi>S</mi><mi>a</mi><mi>t</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>∪</mo><mo>(</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo>)</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></math></span> for every <span><math><mi>q</mi><mo>≥</mo><mi>p</mi><mo>≥</mo><mn>2</mn></math></span> and <span><math><mi>t</mi><mo>≥</mo><mn>2</mn></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 10","pages":"Article 114530"},"PeriodicalIF":0.7,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143817396","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 characterization of graphs with maximum k-clique isolation number","authors":"Siyue Chen,&nbsp;Qing Cui,&nbsp;Lingping Zhong","doi":"10.1016/j.disc.2025.114531","DOIUrl":"10.1016/j.disc.2025.114531","url":null,"abstract":"<div><div>For any positive integer <em>k</em> and any graph <em>G</em>, a subset <em>D</em> of vertices of <em>G</em> is called a <em>k</em>-clique isolating set of <em>G</em> if <span><math><mi>G</mi><mo>−</mo><mi>N</mi><mo>[</mo><mi>D</mi><mo>]</mo></math></span> does not contain <em>k</em>-clique as a subgraph. The <em>k</em>-clique isolation number of <em>G</em>, denoted by <span><math><mi>ι</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span>, is the minimum cardinality of a <em>k</em>-clique isolating set of <em>G</em>. Borg, Fenech and Kaemawichanurat (Discrete Math. 343 (2020) 111879) proved that if <em>G</em> is a connected <em>n</em>-vertex graph, then <span><math><mi>ι</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>k</mi><mo>)</mo><mo>≤</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfrac></math></span> unless <em>G</em> is a <em>k</em>-clique, or <span><math><mi>k</mi><mo>=</mo><mn>2</mn></math></span> and <em>G</em> is a 5-cycle. At the end of their paper, Borg, Fenech and Kaemawichanurat asked for a characterization of all connected <em>n</em>-vertex graphs <em>G</em> such that <span><math><mi>ι</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>k</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfrac></math></span>. An old result of Payan and Xuong, and independently of Fink et al., in the 1980s has already answered this problem for the case <span><math><mi>k</mi><mo>=</mo><mn>1</mn></math></span>. Very recently, the case when <span><math><mi>k</mi><mo>=</mo><mn>2</mn></math></span> was solved by Boyer and Goddard, and the case when <span><math><mi>k</mi><mo>=</mo><mn>3</mn></math></span> was solved by the first two authors of the present paper and Zhang. In this paper, we solve all the remaining cases. We show that except an infinite family of graphs, there are exactly 7 such graphs when <span><math><mi>k</mi><mo>=</mo><mn>4</mn></math></span> and exactly <span><math><mi>k</mi><mo>+</mo><mn>2</mn></math></span> such graphs when <span><math><mi>k</mi><mo>≥</mo><mn>5</mn></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 9","pages":"Article 114531"},"PeriodicalIF":0.7,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143817468","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":"Shortness parameters of polyhedral graphs with few distinct vertex degrees","authors":"On-Hei Solomon Lo ,&nbsp;Jens M. Schmidt ,&nbsp;Nico Van Cleemput ,&nbsp;Carol T. Zamfirescu","doi":"10.1016/j.disc.2025.114518","DOIUrl":"10.1016/j.disc.2025.114518","url":null,"abstract":"<div><div>We devise several new upper bounds for shortness parameters of regular polyhedra and of the polyhedra that have two vertex degrees, and relate these to each other. Grünbaum and Walther showed that quartic polyhedra have shortness exponent at most <span><math><mi>log</mi><mo>⁡</mo><mn>22</mn><mo>/</mo><mi>log</mi><mo>⁡</mo><mn>23</mn></math></span>. This was subsequently improved by Harant to <span><math><mi>log</mi><mo>⁡</mo><mn>16</mn><mo>/</mo><mi>log</mi><mo>⁡</mo><mn>17</mn></math></span>, which holds even when all faces are either triangles or of length <em>k</em>, for infinitely many <em>k</em>. We complement Harant's result by strengthening the Grünbaum-Walther bound to <span><math><mi>log</mi><mo>⁡</mo><mn>4</mn><mo>/</mo><mi>log</mi><mo>⁡</mo><mn>5</mn></math></span>, and showing that this bound even holds for the family of quartic polyhedra with faces of length at most 7. Furthermore, we prove that for every <span><math><mn>4</mn><mo>≤</mo><mi>ℓ</mi><mo>≤</mo><mn>8</mn></math></span> the shortness exponent of the polyhedra having only vertices of degree 3 or <em>ℓ</em> is at most <span><math><mi>log</mi><mo>⁡</mo><mn>5</mn><mo>/</mo><mi>log</mi><mo>⁡</mo><mn>7</mn></math></span>. Motivated by work of Ewald, we show that polyhedral quadrangulations with maximum degree at most 5 have shortness coefficient at most 30/37. Finally, we define path analogues for shortness parameters, and propose first dependencies between these measures.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114518"},"PeriodicalIF":0.7,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143816688","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 safeness condition for minimal separators based on vertex connectivity","authors":"Michel Medema,&nbsp;Alexander Lazovik","doi":"10.1016/j.disc.2025.114524","DOIUrl":"10.1016/j.disc.2025.114524","url":null,"abstract":"<div><div>The treewidth is a measure that quantifies how tree-like a graph is. Its interest stems from the fact that many problems on graphs that are NP-complete for arbitrary graphs become solvable in polynomial time when restricted to graphs with bounded treewidth. Unfortunately, computing the treewidth of a graph is itself an NP-complete problem. A preprocessing technique that has proven to be highly effective at reducing the size of a graph for which the treewidth is to be computed is splitting the graph into multiple smaller subgraphs using separators that are safe for treewidth, allowing the treewidth of the complete graph to be computed as the maximum treewidth over these subgraphs. This paper introduces a new class of safe separators that largely generalises the existing classes of safe separators and opens up the possibility of designing even more powerful preprocessing techniques. An experimental evaluation on a set of synthetic graphs with known safe separators demonstrates that decomposing the graphs using their safe separators has the potential to reduce the execution time of algorithms by up to 10 times in many cases, with a maximum of more than 100 times in the most favourable cases. A heuristic decomposition algorithm is also presented that uses the community structure of a graph to decompose it into subgraphs. The evaluation of this heuristic algorithm on both the synthetic graphs and the graphs from the PACE challenge of 2017 using the same algorithms shows that the potential reduction in execution time is largely maintained while having only a slight impact on the quality as a result of the heuristic-based decomposition.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 9","pages":"Article 114524"},"PeriodicalIF":0.7,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143817469","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":"Non-separating cycles and 5-cycle double covers","authors":"Xiaoyu Li ,&nbsp;Rong-Xia Hao ,&nbsp;Rong Luo ,&nbsp;Cun-Quan Zhang","doi":"10.1016/j.disc.2025.114515","DOIUrl":"10.1016/j.disc.2025.114515","url":null,"abstract":"<div><div>A cycle <em>C</em> in a graph <em>G</em> is non-separating if <span><math><mi>G</mi><mo>−</mo><mi>E</mi><mo>(</mo><mi>C</mi><mo>)</mo></math></span> is connected. As an approach to attack the well-known cycle double cover conjecture and its stronger version: the 5-cycle double cover conjecture, it is conjectured by Hoffmann-Ostenhof (2017) that if a 2-edge connected cubic graph has a non-separating cycle <em>C</em>, then <em>G</em> has a cycle double cover. Hoffmann-Ostenhof <em>et al.</em> (European J. Combin. 2019) show that if a 2-edge connected graph <em>G</em> has a non-separating cycle <em>C</em> such that <span><math><mi>ϵ</mi><mo>(</mo><mi>G</mi><mo>−</mo><mi>E</mi><mo>(</mo><mi>C</mi><mo>)</mo><mo>)</mo><mo>=</mo><mn>0</mn></math></span> and <span><math><mi>w</mi><mo>(</mo><mi>C</mi><mo>)</mo><mo>≤</mo><mn>3</mn></math></span>, where <span><math><mi>ϵ</mi><mo>(</mo><mi>G</mi><mo>−</mo><mi>E</mi><mo>(</mo><mi>C</mi><mo>)</mo><mo>)</mo></math></span> and <span><math><mi>w</mi><mo>(</mo><mi>C</mi><mo>)</mo></math></span> are the rank of the cycle space of <span><math><mi>G</mi><mo>−</mo><mi>E</mi><mo>(</mo><mi>C</mi><mo>)</mo></math></span> and the number of the components of <em>C</em>, respectively, then <em>G</em> has a 5-cycle double cover containing <em>C</em> unless <em>G</em> is contractible to the Petersen graph, in which case, <em>G</em> has a 6-cycle double cover. In this paper we extend their result and prove that if a 2-edge connected graph <em>G</em> has a non-separating cycle <em>C</em> such that <span><math><mo>⌊</mo><mi>ϵ</mi><mo>(</mo><mi>G</mi><mo>−</mo><mi>E</mi><mo>(</mo><mi>C</mi><mo>)</mo><mo>)</mo><mo>+</mo><mfrac><mrow><mn>5</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi>w</mi><mo>(</mo><mi>C</mi><mo>)</mo><mo>⌋</mo><mo>≤</mo><mn>8</mn></math></span>, then <em>G</em> has a 5-cycle double cover or 6-cycle double cover containing <em>C</em> depending on whether <em>G</em> is contractible to the Petersen graph or not. Examples are also constructed in this paper showing the sharpness of the main theorem.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 9","pages":"Article 114515"},"PeriodicalIF":0.7,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143817467","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 saturation number for unions of four cliques","authors":"Ruo-Xuan Li,&nbsp;Rong-Xia Hao,&nbsp;Zhen He,&nbsp;Wen-Han Zhu","doi":"10.1016/j.disc.2025.114532","DOIUrl":"10.1016/j.disc.2025.114532","url":null,"abstract":"<div><div>A graph <em>G</em> is <em>H</em>-saturated if <em>G</em> does not contain a copy of <em>H</em>, but the addition of any edge <span><math><mi>e</mi><mo>∈</mo><mover><mrow><mi>G</mi></mrow><mo>‾</mo></mover></math></span> would create a copy of <em>H</em>. The saturation number <span><math><mi>s</mi><mi>a</mi><mi>t</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> for a graph <em>H</em> is the minimal number of edges in any <em>H</em>-saturated graph of order <em>n</em>. The <span><math><mi>s</mi><mi>a</mi><mi>t</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>∪</mo><msub><mrow><mi>K</mi></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><mo>∪</mo><msub><mrow><mi>K</mi></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></msub><mo>)</mo></math></span> with <span><math><msub><mrow><mi>p</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>≥</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> was determined in [Discrete Math. 347 (2024) 113868]. In this paper, <span><math><mi>s</mi><mi>a</mi><mi>t</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>∪</mo><msub><mrow><mi>K</mi></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><mo>∪</mo><msub><mrow><mi>K</mi></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></msub><mo>∪</mo><msub><mrow><mi>K</mi></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow></msub><mo>)</mo></math></span> with <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≥</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> for <span><math><mn>2</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mn>3</mn></math></span> and <span><math><mn>4</mn><mo>≤</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≤</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> is determined.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 10","pages":"Article 114532"},"PeriodicalIF":0.7,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143808791","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}