Pub Date : 2026-05-01Epub Date: 2025-12-08DOI: 10.1016/j.disc.2025.114914
Ruike Wang, Zhenzhen Lou
Spectral stability theorems have been a crucial aspect of graph theory research. Consider a graph G with size m and spectral radius . Building on the solid foundation laid by previous works in this rich field, this paper presents novel and valuable findings related to stability. Wang and Guo (2024) [16] showed an important result. Given that and , when , then G contains either a quadrilateral or a star of size . In this paper, we take a significant step forward by generalizing this result. Precisely, for and , when , we prove that G contains either a copy of (a complete bipartite graph with two vertices on one side and t vertices on the other side) or a star of size . This generalization contributes to a more profound understanding of the spectral and structural aspects of graphs, as well as their stability properties.
谱稳定性定理一直是图论研究的一个重要方面。考虑一个大小为m，谱半径为ρ(G)的图G。本文在前人在这一丰富领域的工作奠定的坚实基础上，提出了与稳定性有关的新颖而有价值的发现。Wang and Guo(2024)[16]给出了重要的结果。设m=Ω（k4）且k≥0，当ρ(G)≥m - k时，则G包含大小为m - k的四边形或星形。在本文中，我们通过推广这一结果向前迈出了重要的一步。准确地说，对于m=Ω（k4）和2≤t≤k+2，当ρ(G)≥m−k+t−2时，我们证明了G包含K2的一个副本，t（一侧有两个顶点，另一侧有t个顶点的完全二部图）或一个大小为m−k+t−2的星。这种推广有助于更深刻地理解图的谱和结构方面，以及它们的稳定性。
{"title":"A spectral stability result regarding the complete bipartite graph K2,t","authors":"Ruike Wang, Zhenzhen Lou","doi":"10.1016/j.disc.2025.114914","DOIUrl":"10.1016/j.disc.2025.114914","url":null,"abstract":"<div><div>Spectral stability theorems have been a crucial aspect of graph theory research. Consider a graph <em>G</em> with size <em>m</em> and spectral radius <span><math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. Building on the solid foundation laid by previous works in this rich field, this paper presents novel and valuable findings related to stability. Wang and Guo (2024) <span><span>[16]</span></span> showed an important result. Given that <span><math><mi>m</mi><mo>=</mo><mi>Ω</mi><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>)</mo></math></span> and <span><math><mi>k</mi><mo>≥</mo><mn>0</mn></math></span>, when <span><math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><msqrt><mrow><mi>m</mi><mo>−</mo><mi>k</mi></mrow></msqrt></math></span>, then <em>G</em> contains either a quadrilateral or a star of size <span><math><mi>m</mi><mo>−</mo><mi>k</mi></math></span>. In this paper, we take a significant step forward by generalizing this result. Precisely, for <span><math><mi>m</mi><mo>=</mo><mi>Ω</mi><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>)</mo></math></span> and <span><math><mn>2</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mi>k</mi><mo>+</mo><mn>2</mn></math></span>, when <span><math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><msqrt><mrow><mi>m</mi><mo>−</mo><mi>k</mi><mo>+</mo><mi>t</mi><mo>−</mo><mn>2</mn></mrow></msqrt></math></span>, we prove that <em>G</em> contains either a copy of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>t</mi></mrow></msub></math></span> (a complete bipartite graph with two vertices on one side and <em>t</em> vertices on the other side) or a star of size <span><math><mi>m</mi><mo>−</mo><mi>k</mi><mo>+</mo><mi>t</mi><mo>−</mo><mn>2</mn></math></span>. This generalization contributes to a more profound understanding of the spectral and structural aspects of graphs, as well as their stability properties.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 5","pages":"Article 114914"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145697781","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}
Pub Date : 2026-05-01Epub Date: 2025-12-08DOI: 10.1016/j.disc.2025.114915
Hyunsoo Cho , Ho-Hyeong Lee , Kyeongjun Lee , Hayan Nam , Jaebum Sohn
Huang and Wang began counting the number of simultaneous core partitions with the fixed number of corners. They enumerated the number of -core partitions with m corners and -core partitions with m corners. Later, Cho, Huh, and Sohn gave a formula for the number of -core partitions with m corners. In this paper, we show an expression for the number of -core partitions with m corners.
{"title":"Corners of (t,tk ± 1)-core partitions","authors":"Hyunsoo Cho , Ho-Hyeong Lee , Kyeongjun Lee , Hayan Nam , Jaebum Sohn","doi":"10.1016/j.disc.2025.114915","DOIUrl":"10.1016/j.disc.2025.114915","url":null,"abstract":"<div><div>Huang and Wang began counting the number of simultaneous core partitions with the fixed number of corners. They enumerated the number of <span><math><mo>(</mo><mi>t</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-core partitions with <em>m</em> corners and <span><math><mo>(</mo><mi>t</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>t</mi><mo>+</mo><mn>2</mn><mo>)</mo></math></span>-core partitions with <em>m</em> corners. Later, Cho, Huh, and Sohn gave a formula for the number of <span><math><mo>(</mo><mi>t</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>t</mi><mo>+</mo><mi>p</mi><mo>)</mo></math></span>-core partitions with <em>m</em> corners. In this paper, we show an expression for the number of <span><math><mo>(</mo><mi>t</mi><mo>,</mo><mi>t</mi><mi>k</mi><mo>±</mo><mn>1</mn><mo>)</mo></math></span>-core partitions with <em>m</em> corners.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 5","pages":"Article 114915"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145697836","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}
Pub Date : 2026-05-01Epub Date: 2025-12-09DOI: 10.1016/j.disc.2025.114912
Thomas W. Cusick , Younhwan Cheon
<div><div>Rotation symmetric (RS) Boolean functions have been extensively studied for over twenty years because of their applications in cryptography and coding theory. The present paper studies degree 3 RS functions, and relies extensively on the theory of the affine equivalence of such functions developed in <span><span>[3]</span></span>. It is known <span><span>[1]</span></span> that if <span><math><mi>f</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span> is the RS Boolean function in <em>n</em> variables generated by the monomial <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> (notation <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msub></math></span>), then the sequence <span><math><mi>w</mi><mi>t</mi><mo>(</mo><msub><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>,</mo><mspace></mspace><mi>n</mi><mo>=</mo><mi>i</mi><mo>,</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo>,</mo><mo>…</mo></math></span>, where <span><math><mi>w</mi><mi>t</mi><mo>(</mo><msub><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span> denotes the Hamming weight of the function, satisfies a linear recursion with integer coefficients and this recursion can be explicitly computed with a method given in <span><span>[1]</span></span>. It was observed in <span><span>[10, Lemma 3.5, p. 396]</span></span> that the functions <span><math><msub><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msub></math></span> and <span><math><msub><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>5</mn><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msub></math></span> have the same weights for every <em>n</em> even though the two functions are not affine equivalent for infinitely many values of <em>n</em>. It was not clear what the explanation for that is. This paper answers that question and gives a general theory that provides
{"title":"Recursion polynomial for cubic rotation symmetric Boolean functions","authors":"Thomas W. Cusick , Younhwan Cheon","doi":"10.1016/j.disc.2025.114912","DOIUrl":"10.1016/j.disc.2025.114912","url":null,"abstract":"<div><div>Rotation symmetric (RS) Boolean functions have been extensively studied for over twenty years because of their applications in cryptography and coding theory. The present paper studies degree 3 RS functions, and relies extensively on the theory of the affine equivalence of such functions developed in <span><span>[3]</span></span>. It is known <span><span>[1]</span></span> that if <span><math><mi>f</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span> is the RS Boolean function in <em>n</em> variables generated by the monomial <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> (notation <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msub></math></span>), then the sequence <span><math><mi>w</mi><mi>t</mi><mo>(</mo><msub><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>,</mo><mspace></mspace><mi>n</mi><mo>=</mo><mi>i</mi><mo>,</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo>,</mo><mo>…</mo></math></span>, where <span><math><mi>w</mi><mi>t</mi><mo>(</mo><msub><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span> denotes the Hamming weight of the function, satisfies a linear recursion with integer coefficients and this recursion can be explicitly computed with a method given in <span><span>[1]</span></span>. It was observed in <span><span>[10, Lemma 3.5, p. 396]</span></span> that the functions <span><math><msub><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msub></math></span> and <span><math><msub><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>5</mn><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msub></math></span> have the same weights for every <em>n</em> even though the two functions are not affine equivalent for infinitely many values of <em>n</em>. It was not clear what the explanation for that is. This paper answers that question and gives a general theory that provides ","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 5","pages":"Article 114912"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145749819","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}
Pub Date : 2026-05-01Epub Date: 2026-01-12DOI: 10.1016/j.disc.2026.114995
Jun-Jie Huang , Yan-Quan Feng , Jin-Xin Zhou
For and a graph Γ, a sequence of vertices of Γ is called an s-arc if is adjacent to for and for , and an s-geodesic if in addition and has distance s. We say that Γ is s-arc transitive if its automorphism group is transitive on the set of s-arcs, and s-geodesic transitive if for each , Γ has at least one i-geodesic, and its automorphism group is transitive on the set of i-geodesics for all . In Jin (2015) [15], Jin initiated the investigation of tetravalent 3-geodesic transitive but not 3-arc transitive graphs, and he classified all such graphs of girth at most 4 and conjectured that there do not exist such graphs of girth 5. In this paper, a complete classification is given of tetravalent 3-geodesic transitive but not 3-arc transitive graphs. As a result, we find that all such graphs have girth at most 5, and that there exist four such graphs of girth 5. This disproves Jin's conjecture.
{"title":"On tetravalent 3-geodesic transitive graphs","authors":"Jun-Jie Huang , Yan-Quan Feng , Jin-Xin Zhou","doi":"10.1016/j.disc.2026.114995","DOIUrl":"10.1016/j.disc.2026.114995","url":null,"abstract":"<div><div>For <span><math><mi>s</mi><mo>≥</mo><mn>1</mn></math></span> and a graph Γ, a sequence <span><math><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>)</mo></math></span> of vertices of Γ is called an <em>s-arc</em> if <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is adjacent to <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> for <span><math><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>s</mi><mo>−</mo><mn>1</mn></math></span> and <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>≠</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> for <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>s</mi><mo>−</mo><mn>1</mn></math></span>, and an <em>s-geodesic</em> if in addition <span><math><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span> has distance <em>s</em>. We say that Γ is <em>s-arc transitive</em> if its automorphism group is transitive on the set of <em>s</em>-arcs, and <em>s-geodesic transitive</em> if for each <span><math><mi>i</mi><mo>≤</mo><mi>s</mi></math></span>, Γ has at least one <em>i</em>-geodesic, and its automorphism group is transitive on the set of <em>i</em>-geodesics for all <span><math><mi>i</mi><mo>≤</mo><mi>s</mi></math></span>. In Jin (2015) <span><span>[15]</span></span>, Jin initiated the investigation of tetravalent 3-geodesic transitive but not 3-arc transitive graphs, and he classified all such graphs of girth at most 4 and conjectured that there do not exist such graphs of girth 5. In this paper, a complete classification is given of tetravalent 3-geodesic transitive but not 3-arc transitive graphs. As a result, we find that all such graphs have girth at most 5, and that there exist four such graphs of girth 5. This disproves Jin's conjecture.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 5","pages":"Article 114995"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977656","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}
Pub Date : 2026-05-01Epub Date: 2025-12-08DOI: 10.1016/j.disc.2025.114937
Meng Liu , Yusheng Li
For a graph G, a connected graph H is said to be G-good if the Ramsey number is equal to the lower bound , where and are the chromatic number and the chromatic surplus of G, respectively. Let be the book graph that consists of n copies of sharing a common . Recently, avoiding use of the regularity lemma, Fox, He and Wigderson (2023) revisit several Ramsey goodness results involving books. They commented that it would be very interesting to see how far one can push these ideas. In this paper, we describe graphs G with exactly such that are G-good for large n, where the size of a color class of G can be at most δn for some .
{"title":"Ramsey goodness of large books revisited","authors":"Meng Liu , Yusheng Li","doi":"10.1016/j.disc.2025.114937","DOIUrl":"10.1016/j.disc.2025.114937","url":null,"abstract":"<div><div>For a graph <em>G</em>, a connected graph <em>H</em> is said to be <em>G</em>-good if the Ramsey number <span><math><mi>r</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> is equal to the lower bound <span><math><mo>(</mo><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mo>|</mo><mi>H</mi><mo>|</mo><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mi>s</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, where <span><math><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and <span><math><mi>s</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> are the chromatic number and the chromatic surplus of <em>G</em>, respectively. Let <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> be the book graph that consists of <em>n</em> copies of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> sharing a common <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>. Recently, avoiding use of the regularity lemma, Fox, He and Wigderson (2023) revisit several Ramsey goodness results involving books. They commented that it would be very interesting to see how far one can push these ideas. In this paper, we describe graphs <em>G</em> with <span><math><mi>s</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span> exactly such that <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> are <em>G</em>-good for large <em>n</em>, where the size of a color class of <em>G</em> can be at most <em>δn</em> for some <span><math><mi>δ</mi><mo>></mo><mn>0</mn></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 5","pages":"Article 114937"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145697837","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}
Pub Date : 2026-05-01Epub Date: 2025-12-15DOI: 10.1016/j.disc.2025.114917
J.G. Bradley-Thrush
A one-parameter generalization is obtained for the non-terminating version of Bender's generalized q-Vandermonde sum. This formula includes the summation as a special case. Another special case is interpreted combinatorially and given a bijective proof.
{"title":"A generalization of Bender's q-Vandermonde sum","authors":"J.G. Bradley-Thrush","doi":"10.1016/j.disc.2025.114917","DOIUrl":"10.1016/j.disc.2025.114917","url":null,"abstract":"<div><div>A one-parameter generalization is obtained for the non-terminating version of Bender's generalized <em>q</em>-Vandermonde sum. This formula includes the <span><math><mmultiscripts><mrow><mi>ϕ</mi></mrow><mrow><mn>5</mn></mrow><none></none><mprescripts></mprescripts><mrow><mn>6</mn></mrow><none></none></mmultiscripts></math></span> summation as a special case. Another special case is interpreted combinatorially and given a bijective proof.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 5","pages":"Article 114917"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145798327","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}
Pub Date : 2026-05-01Epub Date: 2025-12-15DOI: 10.1016/j.disc.2025.114946
Shuaijun Liu, Shangwei Lin, Lin Yang
The k-restricted arc-connectivity of digraphs, as a generalization of the arc-connectivity, is an important index to measure the reliability of directed networks. An arc subset S of a strongly connected digraph D is a k-restricted arc-cut if has a strong component with order at least k such that contains a connected subdigraph with order k. The k-restricted arc-connectivity of the digraph D is the minimum cardinality over all k-restricted arc-cuts of D. In this paper, we present a degree sum condition for a strongly connected digraph to be optimal in terms of 3-restricted arc-connectivity, and give an example to show that the lower bound on the degree sum in this result is sharp.
{"title":"Degree sum conditions for optimal 3-restricted arc-connected digraphs","authors":"Shuaijun Liu, Shangwei Lin, Lin Yang","doi":"10.1016/j.disc.2025.114946","DOIUrl":"10.1016/j.disc.2025.114946","url":null,"abstract":"<div><div>The <em>k</em>-restricted arc-connectivity of digraphs, as a generalization of the arc-connectivity, is an important index to measure the reliability of directed networks. An arc subset <em>S</em> of a strongly connected digraph <em>D</em> is a <em>k</em>-restricted arc-cut if <span><math><mi>D</mi><mo>−</mo><mi>S</mi></math></span> has a strong component <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> with order at least <em>k</em> such that <span><math><mi>D</mi><mo>−</mo><mi>V</mi><mo>(</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math></span> contains a connected subdigraph with order <em>k</em>. The <em>k</em>-restricted arc-connectivity <span><math><msup><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>(</mo><mi>D</mi><mo>)</mo></math></span> of the digraph <em>D</em> is the minimum cardinality over all <em>k</em>-restricted arc-cuts of <em>D</em>. In this paper, we present a degree sum condition for a strongly connected digraph to be optimal in terms of 3-restricted arc-connectivity, and give an example to show that the lower bound on the degree sum in this result is sharp.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 5","pages":"Article 114946"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145798328","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}
Pub Date : 2026-05-01Epub Date: 2025-12-29DOI: 10.1016/j.disc.2025.114963
Michel Mollard
The Fibonacci cube is the subgraph of the hypercube induced by vertices with no consecutive 1s. We study a one parameter generalization, p-th order Fibonacci cubes , which are subgraphs of induced by strings without p consecutive 1 s. We show the link between vertices of and compositions of integers with parts in . Among other enumerative properties, we study the order, size and cube polynomial of as well as their generating functions. Many of the given expressions are similar to those for Fibonacci cubes, where the p-nomial coefficients play the role of binomial coefficients. We also show that maximal induced hypercubes in Fibonacci p-cubes , another generalization of Fibonacci cubes, are connected to vertices of -th order Fibonacci cubes. We use this link to determine the maximal cube polynomial of Fibonacci p-cubes.
{"title":"p-th order generalized Fibonacci cubes and maximal cubes in Fibonacci p-cubes","authors":"Michel Mollard","doi":"10.1016/j.disc.2025.114963","DOIUrl":"10.1016/j.disc.2025.114963","url":null,"abstract":"<div><div>The Fibonacci cube <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is the subgraph of the hypercube <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> induced by vertices with no consecutive 1s. We study a one parameter generalization, <em>p</em>-th order Fibonacci cubes <span><math><msubsup><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow></msubsup></math></span>, which are subgraphs of <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> induced by strings without <em>p</em> consecutive 1 s. We show the link between vertices of <span><math><msubsup><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow></msubsup></math></span> and compositions of integers with parts in <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>p</mi><mo>}</mo></math></span>. Among other enumerative properties, we study the order, size and cube polynomial of <span><math><msubsup><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow></msubsup></math></span> as well as their generating functions. Many of the given expressions are similar to those for Fibonacci cubes, where the <em>p</em>-nomial coefficients play the role of binomial coefficients. We also show that maximal induced hypercubes in Fibonacci <em>p</em>-cubes <span><math><msubsup><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>p</mi></mrow></msubsup></math></span>, another generalization of Fibonacci cubes, are connected to vertices of <span><math><mi>p</mi><mo>+</mo><mn>1</mn></math></span>-th order Fibonacci cubes. We use this link to determine the maximal cube polynomial of Fibonacci <em>p</em>-cubes.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 5","pages":"Article 114963"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145884984","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}
Pub Date : 2026-05-01Epub Date: 2025-12-23DOI: 10.1016/j.disc.2025.114960
Long Wang, Xianya Geng
The multiplicity of an adjacency eigenvalue λ of a graph G is denoted by . In a connected graph G with at least two vertices, a vertex v is called external if is connected. Let be the number of external vertices in G. Recently, Wong et al. ([11]) proved that for any , and characterized the extremal graphs G with for the special case when . They also posed the following problem: If a connected graph G has an eigenvalue λ such that, how about the structure of G? and how about the possible value of λ? In the present paper, we give a complete solution for this problem.
图G的邻接特征值λ的多重度用m（G,λ）表示。在至少有两个顶点的连通图G中，如果G−v连通，则顶点v称为外部。最近，Wong et al.（[11]）证明了对于任意λ∈R， m(G,λ)≤λ (G)−1，并刻画了当λ=−1的特殊情况下，m(G,λ)= λ (G)−1的极值图G。他们还提出了以下问题：如果连通图G有一个特征值λ使得m(G,λ)= λ (G)−1，那么G的结构又是怎样的呢？那么λ的可能值呢？在本文中，我们给出了一个完整的解决方案。
{"title":"Characterization of graphs with eigenvalue multiplicity one less than the number of external vertices","authors":"Long Wang, Xianya Geng","doi":"10.1016/j.disc.2025.114960","DOIUrl":"10.1016/j.disc.2025.114960","url":null,"abstract":"<div><div>The multiplicity of an adjacency eigenvalue <em>λ</em> of a graph <em>G</em> is denoted by <span><math><mi>m</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math></span>. In a connected graph <em>G</em> with at least two vertices, a vertex <em>v</em> is called external if <span><math><mi>G</mi><mo>−</mo><mi>v</mi></math></span> is connected. Let <span><math><mi>ϵ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the number of external vertices in <em>G</em>. Recently, Wong et al. (<span><span>[11]</span></span>) proved that <span><math><mi>m</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>λ</mi><mo>)</mo><mo>≤</mo><mi>ϵ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span> for any <span><math><mi>λ</mi><mo>∈</mo><mi>R</mi></math></span>, and characterized the extremal graphs <em>G</em> with <span><math><mi>m</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>λ</mi><mo>)</mo><mo>=</mo><mi>ϵ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span> for the special case when <span><math><mi>λ</mi><mo>=</mo><mo>−</mo><mn>1</mn></math></span>. They also posed the following problem: <em>If a connected graph G has an eigenvalue λ such that</em> <span><math><mi>m</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>λ</mi><mo>)</mo><mo>=</mo><mi>ϵ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span><em>, how about the structure of G? and how about the possible value of λ?</em> In the present paper, we give a complete solution for this problem.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 5","pages":"Article 114960"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145841310","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}
Pub Date : 2026-05-01Epub Date: 2025-12-11DOI: 10.1016/j.disc.2025.114924
Sara Davies , Peter Gill , Daniel Horsley
For positive integers and n, the Zarankiewicz number is the maximum number of edges in a subgraph of that has no complete bipartite subgraph containing s vertices in the part of size m and t vertices in the part of size n. The best general upper bound on Zarankiewicz numbers is a bound due to Roman that can be viewed as the optimal value of a simple linear program. Here we show that in many cases this bound can be improved by adding additional constraints to this linear program. This allows us to prove new upper bounds on Zarankiewicz numbers for many small parameter sets. We are also able to establish a new family of closed form upper bounds on that captures much, but not all, of the power of the new constraints. This bound generalises a recent result of Chen, Horsley and Mammoliti that applied only in the case .
{"title":"Improved upper bounds on Zarankiewicz numbers","authors":"Sara Davies , Peter Gill , Daniel Horsley","doi":"10.1016/j.disc.2025.114924","DOIUrl":"10.1016/j.disc.2025.114924","url":null,"abstract":"<div><div>For positive integers <span><math><mi>s</mi><mo>,</mo><mi>t</mi><mo>,</mo><mi>m</mi></math></span> and <em>n</em>, the Zarankiewicz number <span><math><mi>z</mi><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>;</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> is the maximum number of edges in a subgraph of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span> that has no complete bipartite subgraph containing <em>s</em> vertices in the part of size <em>m</em> and <em>t</em> vertices in the part of size <em>n</em>. The best general upper bound on Zarankiewicz numbers is a bound due to Roman that can be viewed as the optimal value of a simple linear program. Here we show that in many cases this bound can be improved by adding additional constraints to this linear program. This allows us to prove new upper bounds on Zarankiewicz numbers for many small parameter sets. We are also able to establish a new family of closed form upper bounds on <span><math><mi>z</mi><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>;</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> that captures much, but not all, of the power of the new constraints. This bound generalises a recent result of Chen, Horsley and Mammoliti that applied only in the case <span><math><mi>s</mi><mo>=</mo><mn>2</mn></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 5","pages":"Article 114924"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145749336","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}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号