{"title":"On {1,2}-distance-balancedness of generalized Petersen graphs","authors":"Gang Ma , Jianfeng Wang , Sandi Klavžar","doi":"10.1016/j.disc.2025.114579","DOIUrl":"10.1016/j.disc.2025.114579","url":null,"abstract":"<div><div>A connected graph <em>G</em> of diameter <span><math><mrow><mi>diam</mi></mrow><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mi>ℓ</mi></math></span> is <em>ℓ</em>-distance-balanced if <span><math><mo>|</mo><msub><mrow><mi>W</mi></mrow><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>|</mo><mo>=</mo><mo>|</mo><msub><mrow><mi>W</mi></mrow><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>|</mo></math></span> for every <span><math><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> with <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>=</mo><mi>ℓ</mi></math></span>, where <span><math><msub><mrow><mi>W</mi></mrow><mrow><mi>x</mi><mi>y</mi></mrow></msub></math></span> is the set of vertices of <em>G</em> that are closer to <em>x</em> than to <em>y</em>. It is proved that if <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span> and <span><math><mi>n</mi><mo>></mo><mi>k</mi><mo>(</mo><mi>k</mi><mo>+</mo><mn>2</mn><mo>)</mo></math></span>, then the generalized Petersen graph <span><math><mi>G</mi><mi>P</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> is not distance-balanced and that <span><math><mi>G</mi><mi>P</mi><mo>(</mo><mi>k</mi><mo>(</mo><mi>k</mi><mo>+</mo><mn>2</mn><mo>)</mo><mo>,</mo><mi>k</mi><mo>)</mo></math></span> is distance-balanced. It is also proved that if <span><math><mi>k</mi><mo>≥</mo><mn>6</mn></math></span> where <em>k</em> is even, and <span><math><mi>n</mi><mo>></mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msup><mrow><mi>k</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>−</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>9</mn><mi>k</mi></math></span>, or if <span><math><mi>k</mi><mo>≥</mo><mn>5</mn></math></span> where <em>k</em> is odd, and <span><math><mi>n</mi><mo>></mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msup><mrow><mi>k</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mfrac><mrow><mn>31</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mi>k</mi></math></span>, then <span><math><mi>G</mi><mi>P</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> is not 2-distance-balanced.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 10","pages":"Article 114579"},"PeriodicalIF":0.7,"publicationDate":"2025-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144105985","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":"Covering the edges of r-graphs with perfect matchings","authors":"Olha Silina","doi":"10.1016/j.disc.2025.114581","DOIUrl":"10.1016/j.disc.2025.114581","url":null,"abstract":"<div><div>An <em>r</em>-graph is an <em>r</em>-regular graph with no odd cut of size less than <em>r</em>. A well-celebrated result due to Lovász says that for such graphs the linear system <span><math><mi>A</mi><mi>x</mi><mo>=</mo><mn>1</mn></math></span> has a solution in <span><math><mi>Z</mi><mo>/</mo><mn>2</mn></math></span>, where <em>A</em> is the <span><math><mn>0</mn><mo>,</mo><mn>1</mn></math></span> edge to perfect matching incidence matrix. Note that we allow <em>x</em> to have negative entries. In this paper, we present an improved version of Lovász's result, proving that, in fact, there is a solution <em>x</em> with all entries being either integer or <span><math><mo>+</mo><mn>1</mn><mo>/</mo><mn>2</mn></math></span> and corresponding to a linearly independent set of perfect matchings. Moreover, the total number of <span><math><mo>+</mo><mn>1</mn><mo>/</mo><mn>2</mn></math></span>'s is at most 6<em>k</em>, where k is the number of Petersen bricks in the tight cut decomposition of the graph.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 10","pages":"Article 114581"},"PeriodicalIF":0.7,"publicationDate":"2025-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144099507","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 planar Turán number of double star S2,4","authors":"Xin Xu, Jiawei Shao","doi":"10.1016/j.disc.2025.114571","DOIUrl":"10.1016/j.disc.2025.114571","url":null,"abstract":"<div><div>Planar Turán number <span><math><msub><mrow><mi>ex</mi></mrow><mrow><mi>P</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> of <em>H</em> is the maximum number of edges in an <em>n</em>-vertex planar graph which does not contain <em>H</em> as a subgraph. Ghosh, Győri, Paulos and Xiao initiated the topic of the planar Turán number for double stars. In this paper, we prove that <span><math><msub><mrow><mi>ex</mi></mrow><mrow><mi>P</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>2</mn><mo>,</mo><mn>4</mn></mrow></msub><mo>)</mo><mo>≤</mo><mfrac><mrow><mn>31</mn></mrow><mrow><mn>14</mn></mrow></mfrac><mi>n</mi></math></span> for <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span>, and show that equality holds for infinitely many integers <em>n</em>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 11","pages":"Article 114571"},"PeriodicalIF":0.7,"publicationDate":"2025-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144088712","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":"Properly colored even cycles in edge-colored complete balanced bipartite graphs","authors":"Shanshan Guo , Fei Huang , Jinjiang Yuan , C.T. Ng , T.C.E. Cheng","doi":"10.1016/j.disc.2025.114575","DOIUrl":"10.1016/j.disc.2025.114575","url":null,"abstract":"<div><div>Consider a complete balanced bipartite graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span> and let <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi></mrow><mrow><mi>c</mi></mrow></msubsup></math></span> be an edge-colored version of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span> that is obtained from <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span> by having each edge assigned a certain color. A subgraph <em>H</em> of <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi></mrow><mrow><mi>c</mi></mrow></msubsup></math></span> is called properly colored (PC) if every two adjacent edges of <em>H</em> have distinct colors. <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi></mrow><mrow><mi>c</mi></mrow></msubsup></math></span> is called properly vertex-even-pancyclic if for every vertex <span><math><mi>u</mi><mo>∈</mo><mi>V</mi><mo>(</mo><msubsup><mrow><mi>K</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi></mrow><mrow><mi>c</mi></mrow></msubsup><mo>)</mo></math></span> and for every even integer <em>k</em> with <span><math><mn>4</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mn>2</mn><mi>n</mi></math></span>, there exists a PC <em>k</em>-cycle containing <em>u</em>. The minimum color degree <span><math><msup><mrow><mi>δ</mi></mrow><mrow><mi>c</mi></mrow></msup><mo>(</mo><msubsup><mrow><mi>K</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi></mrow><mrow><mi>c</mi></mrow></msubsup><mo>)</mo></math></span> of <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi></mrow><mrow><mi>c</mi></mrow></msubsup></math></span> is the largest integer <em>k</em> such that for every vertex <em>v</em>, there are at least <em>k</em> distinct colors on the edges incident to <em>v</em>. In this paper we study the existence of PC even cycles in <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi></mrow><mrow><mi>c</mi></mrow></msubsup></math></span>. We first show that, for every integer <span><math><mi>t</mi><mo>≥</mo><mn>3</mn></math></span>, every <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi></mrow><mrow><mi>c</mi></mrow></msubsup></math></span> with <span><math><msup><mrow><mi>δ</mi></mrow><mrow><mi>c</mi></mrow></msup><mo>(</mo><msubsup><mrow><mi>K</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi></mrow><mrow><mi>c</mi></mrow></msubsup><mo>)</mo><mo>≥</mo><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mrow><mn>3</mn></mrow></mfrac><mo>+</mo><mi>t</mi></math></span> contains a PC 2-factor <em>H</em> such that every cycle of <em>H</em> has a length of at least <em>t</em>. By using the probabilistic method and absorbing technique, we use the above result to further show that, for every <span><math><mi>ε</mi><mo>></mo><mn>0</mn></math></","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 11","pages":"Article 114575"},"PeriodicalIF":0.7,"publicationDate":"2025-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144088713","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":"112-designs based on attenuated spaces of finite classical polar spaces","authors":"Jing Zhou, Changli Ma, Liwei Zeng","doi":"10.1016/j.disc.2025.114567","DOIUrl":"10.1016/j.disc.2025.114567","url":null,"abstract":"<div><div>In this paper, a class of <span><math><mn>1</mn><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>-designs is constructed based on attenuated spaces of finite classical polar spaces, which generalizes the previous construction in finite classical polar spaces.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 10","pages":"Article 114567"},"PeriodicalIF":0.7,"publicationDate":"2025-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144089252","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}
Rhys J. Evans , Alexander L. Gavrilyuk , Sergey Goryainov , Konstantin Vorob'ev
{"title":"Equitable 2-partitions of the Johnson graphs J(n,3)","authors":"Rhys J. Evans , Alexander L. Gavrilyuk , Sergey Goryainov , Konstantin Vorob'ev","doi":"10.1016/j.disc.2025.114565","DOIUrl":"10.1016/j.disc.2025.114565","url":null,"abstract":"<div><div>We finish the classification of equitable 2-partitions of the Johnson graphs of diameter 3.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 10","pages":"Article 114565"},"PeriodicalIF":0.7,"publicationDate":"2025-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144084511","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}
Shaoshi Chen , Yang Li , Zhicong Lin , Sherry H.F. Yan
{"title":"Bijections around Springer numbers","authors":"Shaoshi Chen , Yang Li , Zhicong Lin , Sherry H.F. Yan","doi":"10.1016/j.disc.2025.114570","DOIUrl":"10.1016/j.disc.2025.114570","url":null,"abstract":"<div><div>Arnol'd proved in 1992 that Springer numbers enumerate the snakes, which are type <em>B</em> analogs of alternating permutations. Chen, Fan and Jia in 2011 introduced the labeled ballot paths and established a “hard” bijection with snakes. Callan conjectured in 2012 and Han–Kitaev–Zhang proved recently that rc-invariant alternating permutations are counted by Springer numbers. Very recently, Chen–Fang–Kitaev–Zhang investigated multi-dimensional permutations and proved that weakly increasing 3-dimensional permutations are also counted by Springer numbers. In this work, we construct a sequence of “natural” bijections linking the above four combinatorial objects.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 11","pages":"Article 114570"},"PeriodicalIF":0.7,"publicationDate":"2025-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144069677","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":"Pattern-restricted permutations of small order","authors":"Kassie Archer, Robert P. Laudone","doi":"10.1016/j.disc.2025.114566","DOIUrl":"10.1016/j.disc.2025.114566","url":null,"abstract":"<div><div>In this paper, we develop a framework for studying pattern avoidance in permutations composed of small cycles via Dyck words, arc diagrams and free blocks. We then apply this to enumerate 132-avoiding permutations of order 3 in terms of the Catalan and Motzkin generating functions, answering a question of Bóna and Smith from 2019. We also enumerate 231-avoiding permutations that are composed only of 3-cycles, 2-cycles, and fixed points.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 10","pages":"Article 114566"},"PeriodicalIF":0.7,"publicationDate":"2025-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144068464","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 minimum size and maximum diameter of an edge-pancyclic graph of a given order","authors":"Chengli Li, Feng Liu, Xingzhi Zhan","doi":"10.1016/j.disc.2025.114576","DOIUrl":"10.1016/j.disc.2025.114576","url":null,"abstract":"<div><div>A <em>k</em>-cycle in a graph is a cycle of length <em>k</em>. A graph <em>G</em> of order <em>n</em> is called edge-pancyclic if for every integer <em>k</em> with <span><math><mn>3</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></math></span>, every edge of <em>G</em> lies in a <em>k</em>-cycle. We give lower and upper bounds on the minimum size of a simple edge-pancyclic graph of a given order, and determine the maximum diameter of such a graph. In the 3-connected case, the precise minimum size is determined. We also determine the minimum size of a graph of a given order with connectivity conditions in which every edge lies in a triangle.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 11","pages":"Article 114576"},"PeriodicalIF":0.7,"publicationDate":"2025-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144069678","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":"Vertex degree sums for perfect matchings in 3-uniform hypergraphs","authors":"Yan Wang , Yi Zhang","doi":"10.1016/j.disc.2025.114564","DOIUrl":"10.1016/j.disc.2025.114564","url":null,"abstract":"<div><div>Let <span><math><mi>n</mi><mo>≡</mo><mn>0</mn><mspace></mspace><mo>(</mo><mspace></mspace><mtext>mod </mtext><mn>3</mn><mspace></mspace><mo>)</mo></math></span> and <span><math><msubsup><mrow><mi>H</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi><mo>/</mo><mn>3</mn></mrow><mrow><mn>2</mn></mrow></msubsup></math></span> be the 3-graph of order <em>n</em>, whose vertex set is partitioned into two sets <em>S</em> and <em>T</em> of size <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mi>n</mi><mo>+</mo><mn>1</mn></math></span> and <span><math><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mi>n</mi><mo>−</mo><mn>1</mn></math></span>, respectively, and whose edge set consists of all triples with at least 2 vertices in <em>T</em>. Suppose that <em>n</em> is sufficiently large and <em>H</em> is a 3-uniform hypergraph of order <em>n</em> with no isolated vertex. Zhang and Lu [Discrete Math. 341 (2018), 748–758] conjectured that if <span><math><mi>deg</mi><mo></mo><mo>(</mo><mi>u</mi><mo>)</mo><mo>+</mo><mi>deg</mi><mo></mo><mo>(</mo><mi>v</mi><mo>)</mo><mo>></mo><mn>2</mn><mo>(</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mn>2</mn><mi>n</mi><mo>/</mo><mn>3</mn></mrow></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow><mo>)</mo></math></span> for any two vertices <em>u</em> and <em>v</em> that are contained in some edge of <em>H</em>, then <em>H</em> contains a perfect matching or <em>H</em> is a subgraph of <span><math><msubsup><mrow><mi>H</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi><mo>/</mo><mn>3</mn></mrow><mrow><mn>2</mn></mrow></msubsup></math></span>. We construct a counter-example to the conjecture. Furthermore, for all <span><math><mi>γ</mi><mo>></mo><mn>0</mn></math></span> and <span><math><mi>n</mi><mo>∈</mo><mn>3</mn><mi>Z</mi></math></span> sufficiently large, we prove that if <span><math><mi>deg</mi><mo></mo><mo>(</mo><mi>u</mi><mo>)</mo><mo>+</mo><mi>deg</mi><mo></mo><mo>(</mo><mi>v</mi><mo>)</mo><mo>></mo><mo>(</mo><mn>3</mn><mo>/</mo><mn>5</mn><mo>+</mo><mi>γ</mi><mo>)</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> for any two vertices <em>u</em> and <em>v</em> that are contained in some edge of <em>H</em>, then <em>H</em> contains a perfect matching or <em>H</em> is a subgraph of <span><math><msubsup><mrow><mi>H</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi><mo>/</mo><mn>3</mn></mrow><mrow><mn>2</mn></mrow></msubsup></math></span>. This implies a result of Zhang, Zhao and Lu [Electron. J. Combin. 25 (3), 2018].</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 10","pages":"Article 114564"},"PeriodicalIF":0.7,"publicationDate":"2025-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144068463","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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