Pub Date : 2026-05-01Epub Date: 2025-12-11DOI: 10.1016/j.disc.2025.114944
Yibo Li , Fengming Dong , Xiaolan Hu , Huiqing Liu
For a connected graph G, a spanning tree T of G is called a homeomorphically irreducible spanning tree (HIST) if T has no vertices of degree 2. In this paper, we show that if G is a graph of order and holds for every pair of non-adjacent vertices u and v in G, then G has a HIST, unless G belongs to three exceptional families of graphs or G has a cut-vertex of degree 2. This result improves the latest conclusion, due to Ito and Tsuchiya, that the existence of a HIST in G can be guaranteed if holds for every pair of non-adjacent vertices u and v in G.
对于连通图G，如果T没有2次顶点，则G的生成树T称为同胚不可约生成树（HIST）。在本文中，我们证明了如果G是一个阶n≥270且| n (u)∪n (v)|≥n−12的图，对于G中每一对不相邻的顶点u和v都成立，那么G有一个HIST，除非G属于三个例外的图族或G有一个2次的切顶点。该结果改进了Ito和Tsuchiya的最新结论，即对于G中的每一对非相邻顶点u和v，如果d(u)+d(v)≥n−1成立，则G中存在HIST。
{"title":"A neighborhood union condition for the existence of a spanning tree without degree 2 vertices","authors":"Yibo Li , Fengming Dong , Xiaolan Hu , Huiqing Liu","doi":"10.1016/j.disc.2025.114944","DOIUrl":"10.1016/j.disc.2025.114944","url":null,"abstract":"<div><div>For a connected graph <em>G</em>, a spanning tree <em>T</em> of <em>G</em> is called a homeomorphically irreducible spanning tree (HIST) if <em>T</em> has no vertices of degree 2. In this paper, we show that if <em>G</em> is a graph of order <span><math><mi>n</mi><mo>≥</mo><mn>270</mn></math></span> and <span><math><mo>|</mo><mi>N</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>∪</mo><mi>N</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>|</mo><mo>≥</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> holds for every pair of non-adjacent vertices <em>u</em> and <em>v</em> in <em>G</em>, then <em>G</em> has a HIST, unless <em>G</em> belongs to three exceptional families of graphs or <em>G</em> has a cut-vertex of degree 2. This result improves the latest conclusion, due to Ito and Tsuchiya, that the existence of a HIST in <em>G</em> can be guaranteed if <span><math><mi>d</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>+</mo><mi>d</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>≥</mo><mi>n</mi><mo>−</mo><mn>1</mn></math></span> holds for every pair of non-adjacent vertices <em>u</em> and <em>v</em> in <em>G</em>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 5","pages":"Article 114944"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145749337","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.2025.114943
David S. Herscovici
We imagine a distribution of pebbles on the vertices of a connected graph. Chung defined a pebbling move as the removal of two pebbles from some vertex and the addition of a pebble to an adjacent vertex. Then the pebbling number of a graph G is the smallest number such that every distribution of pebbles allows one pebble to reach any specified, but arbitrary vertex. Graham conjectured that for all connected graphs G and H. We show that the pebbling number of satisfies for any connected graph G that satisfies the odd two-pebbling property. In particular, .
{"title":"Pebbling Cartesian products of C5","authors":"David S. Herscovici","doi":"10.1016/j.disc.2025.114943","DOIUrl":"10.1016/j.disc.2025.114943","url":null,"abstract":"<div><div>We imagine a distribution of pebbles on the vertices of a connected graph. Chung defined a <em>pebbling move</em> as the removal of two pebbles from some vertex and the addition of a pebble to an adjacent vertex. Then the <em>pebbling number</em> of a graph <em>G</em> is the smallest number <span><math><mi>π</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> such that every distribution of <span><math><mi>π</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> pebbles allows one pebble to reach any specified, but arbitrary vertex. Graham conjectured that <span><math><mi>π</mi><mo>(</mo><mi>G</mi><mo>□</mo><mi>H</mi><mo>)</mo><mo>≤</mo><mi>π</mi><mo>(</mo><mi>G</mi><mo>)</mo><mi>π</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> for all connected graphs <em>G</em> and <em>H</em>. We show that the pebbling number of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>□</mo><mi>G</mi></math></span> satisfies <span><math><mi>π</mi><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>□</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mn>5</mn><mi>π</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> for any connected graph <em>G</em> that satisfies the odd two-pebbling property. In particular, <span><math><mi>π</mi><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>□</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>□</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>)</mo><mo>=</mo><mn>125</mn></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 5","pages":"Article 114943"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145798326","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: 2026-01-02DOI: 10.1016/j.disc.2025.114969
Jun Hu , Gengsheng Zhang , Bo Hou
The concept of the nucleus of a distance-regular graph was recently introduced by P. Terwilliger. Let Γ be a Q-polynomial distance-regular graph with vertex set Y. Let be the Terwilliger algebra of Γ with respect to a fixed vertex . Then the nucleus of Γ with respect to x is a certain T-module. In this paper, we describe the nucleus of the Hamming graph and construct two bases for the nucleus by using the Hamming semilattice . Our main result partially answers an open problem proposed by P. Terwilliger (2025) [21].
{"title":"The nucleus of the Hamming graph H(D,q)","authors":"Jun Hu , Gengsheng Zhang , Bo Hou","doi":"10.1016/j.disc.2025.114969","DOIUrl":"10.1016/j.disc.2025.114969","url":null,"abstract":"<div><div>The concept of the nucleus of a distance-regular graph was recently introduced by P. Terwilliger. Let Γ be a <em>Q</em>-polynomial distance-regular graph with vertex set <em>Y</em>. Let <span><math><mi>T</mi><mo>=</mo><mi>T</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> be the Terwilliger algebra of Γ with respect to a fixed vertex <span><math><mi>x</mi><mo>∈</mo><mi>Y</mi></math></span>. Then the nucleus of Γ with respect to <em>x</em> is a certain <em>T</em>-module. In this paper, we describe the nucleus of the Hamming graph <span><math><mi>H</mi><mo>(</mo><mi>D</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span> and construct two bases for the nucleus by using the Hamming semilattice <span><math><mi>H</mi><mo>(</mo><mi>D</mi><mo>,</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>. Our main result partially answers an open problem proposed by P. Terwilliger (2025) <span><span>[21]</span></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 5","pages":"Article 114969"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145885057","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: 2026-01-06DOI: 10.1016/j.disc.2025.114966
Mariusz Meszka
We complete a solution to the Hamilton-Waterloo problem in the case when 2-factors are either triangle-factors or Hamiltonian cycles. Namely, we prove that for each and r such that , there exists a 2-factorization of the complete graph in which r of the 2-factors are Hamiltonian cycles and the remaining 2-factors are Δ-factors, except when .
{"title":"The Hamilton–Waterloo problem for triangle-factors and Hamiltonian cycles solved","authors":"Mariusz Meszka","doi":"10.1016/j.disc.2025.114966","DOIUrl":"10.1016/j.disc.2025.114966","url":null,"abstract":"<div><div>We complete a solution to the Hamilton-Waterloo problem in the case when 2-factors are either triangle-factors or Hamiltonian cycles. Namely, we prove that for each <span><math><mi>k</mi><mo>≥</mo><mn>1</mn></math></span> and <em>r</em> such that <span><math><mn>0</mn><mo>≤</mo><mi>r</mi><mo>≤</mo><mn>3</mn><mi>k</mi><mo>+</mo><mn>1</mn></math></span>, there exists a 2-factorization of the complete graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>6</mn><mi>k</mi><mo>+</mo><mn>3</mn></mrow></msub></math></span> in which <em>r</em> of the 2-factors are Hamiltonian cycles and the remaining <span><math><mo>(</mo><mn>3</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo>−</mo><mi>r</mi><mo>)</mo></math></span> 2-factors are Δ-factors, except when <span><math><mi>k</mi><mo>=</mo><mi>r</mi><mo>=</mo><mn>1</mn></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 5","pages":"Article 114966"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145939115","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.2025.114916
Wenxia Qu , Wenston J.T. Zang
In this paper, we present the bilateral truncated identity of the quintuple product identity, which is a generalization of the truncated quintuple product identities given by Chan et al. (2016) [6]. Additionally, we provide the bilateral truncated forms of two q-series identities, which are well-known consequences of the quintuple product identity.
{"title":"Bilateral truncated quintuple product identity","authors":"Wenxia Qu , Wenston J.T. Zang","doi":"10.1016/j.disc.2025.114916","DOIUrl":"10.1016/j.disc.2025.114916","url":null,"abstract":"<div><div>In this paper, we present the bilateral truncated identity of the quintuple product identity, which is a generalization of the truncated quintuple product identities given by Chan et al. (2016) <span><span>[6]</span></span>. Additionally, we provide the bilateral truncated forms of two <em>q</em>-series identities, which are well-known consequences of the quintuple product identity.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 5","pages":"Article 114916"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145697835","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.2025.114913
William J. Wesley
We prove new bounds for Ramsey numbers for book graphs . In particular, we show that for an infinite family of n using a block-circulant construction similar to Paley graphs. We obtain improved bounds for several other values of using different block-circulant graphs from SAT and integer programming (IP) solvers. Finally, we enumerate the number of critical graphs for for small r and s using SAT modulo symmetries (SMS).
{"title":"Lower bounds for book Ramsey numbers","authors":"William J. Wesley","doi":"10.1016/j.disc.2025.114913","DOIUrl":"10.1016/j.disc.2025.114913","url":null,"abstract":"<div><div>We prove new bounds for Ramsey numbers for book graphs <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. In particular, we show that <span><math><mi>R</mi><mo>(</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>=</mo><mn>4</mn><mi>n</mi><mo>−</mo><mn>1</mn></math></span> for an infinite family of <em>n</em> using a block-circulant construction similar to Paley graphs. We obtain improved bounds for several other values of <span><math><mi>R</mi><mo>(</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>)</mo></math></span> using different block-circulant graphs from SAT and integer programming (IP) solvers. Finally, we enumerate the number of critical graphs for <span><math><mi>R</mi><mo>(</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>)</mo></math></span> for small <em>r</em> and <em>s</em> using SAT modulo symmetries (SMS).</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 5","pages":"Article 114913"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145697834","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.2025.114922
Vinayak Gupta
Let T be a tree with p pendant vertices, and let denote the multiplicity of the eigenvalue λ of the Laplacian matrix (T). It has recently been shown that if and only if T has p pendant vertices and the distance between any two distinct pendant vertices u and v satisfies . This article provides a complete characterization of all trees T for which .
{"title":"Trees with one as Laplacian eigenvalue with multiplicity two less than the number of pendant vertices","authors":"Vinayak Gupta","doi":"10.1016/j.disc.2025.114922","DOIUrl":"10.1016/j.disc.2025.114922","url":null,"abstract":"<div><div>Let <em>T</em> be a tree with <em>p</em> pendant vertices, and let <span><math><mi>m</mi><mo>(</mo><mi>T</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math></span> denote the multiplicity of the eigenvalue <em>λ</em> of the Laplacian matrix (<em>T</em>). It has recently been shown that <span><math><mi>m</mi><mo>(</mo><mi>T</mi><mo>,</mo><mn>1</mn><mo>)</mo><mo>=</mo><mi>p</mi><mo>−</mo><mn>1</mn></math></span> if and only if <em>T</em> has <em>p</em> pendant vertices and the distance between any two distinct pendant vertices <em>u</em> and <em>v</em> satisfies <span><math><mi>d</mi><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo><mo>≡</mo><mn>2</mn><mspace></mspace><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mn>3</mn></math></span>. This article provides a complete characterization of all trees <em>T</em> for which <span><math><mi>m</mi><mo>(</mo><mi>T</mi><mo>,</mo><mn>1</mn><mo>)</mo><mo>=</mo><mi>p</mi><mo>−</mo><mn>2</mn></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 5","pages":"Article 114922"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145749335","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.2025.114942
Cristian M. Conde , Ezequiel Dratman , Luciano N. Grippo
A signed graph is one that features two types of edges: positive and negative. Balanced signed graphs are those in which all cycles contain an even number of negative edges. In the adjacency matrix of a signed graph, entries can be 0, −1, or 1, depending on whether ij represents no edge, a negative edge, or a positive edge, respectively. The index of the adjacency matrix of a signed graph is less or equal to the index of the adjacency matrix of its underlying graph G, i.e., . Indeed, if is balanced, then . This inequality becomes strict when is an unbalanced signed graph. Recently, Brunetti and Stanić found the whole list of unbalanced signed graphs on n vertices with maximum (resp. minimum) spectral radius. To our knowledge, there has been little research on this problem when unbalanced signed graphs are confined to specific graph classes. In this article, we demonstrate that there is only one unbalanced signed bipartite graph on n vertices with maximum spectral radius, up to an operation on the signed edges known as switching. Additionally, we investigate unbalanced signed complete bipartite graphs on n vertices with a bounded number of edges and maximum spectral radius, where the negative edges induce a tree.
{"title":"On the spectral radius of unbalanced signed bipartite graphs","authors":"Cristian M. Conde , Ezequiel Dratman , Luciano N. Grippo","doi":"10.1016/j.disc.2025.114942","DOIUrl":"10.1016/j.disc.2025.114942","url":null,"abstract":"<div><div>A signed graph is one that features two types of edges: positive and negative. Balanced signed graphs are those in which all cycles contain an even number of negative edges. In the adjacency matrix of a signed graph, entries can be 0, −1, or 1, depending on whether <em>ij</em> represents no edge, a negative edge, or a positive edge, respectively. The index of the adjacency matrix of a signed graph <span><math><mover><mrow><mi>G</mi></mrow><mrow><mo>˙</mo></mrow></mover></math></span> is less or equal to the index of the adjacency matrix of its underlying graph <em>G</em>, i.e., <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mover><mrow><mi>G</mi></mrow><mrow><mo>˙</mo></mrow></mover><mo>)</mo><mo>≤</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. Indeed, if <span><math><mover><mrow><mi>G</mi></mrow><mrow><mo>˙</mo></mrow></mover></math></span> is balanced, then <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mover><mrow><mi>G</mi></mrow><mrow><mo>˙</mo></mrow></mover><mo>)</mo><mo>=</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. This inequality becomes strict when <span><math><mover><mrow><mi>G</mi></mrow><mrow><mo>˙</mo></mrow></mover></math></span> is an unbalanced signed graph. Recently, Brunetti and Stanić found the whole list of unbalanced signed graphs on <em>n</em> vertices with maximum (resp. minimum) spectral radius. To our knowledge, there has been little research on this problem when unbalanced signed graphs are confined to specific graph classes. In this article, we demonstrate that there is only one unbalanced signed bipartite graph on <em>n</em> vertices with maximum spectral radius, up to an operation on the signed edges known as switching. Additionally, we investigate unbalanced signed complete bipartite graphs on <em>n</em> vertices with a bounded number of edges and maximum spectral radius, where the negative edges induce a tree.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 5","pages":"Article 114942"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145749820","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: 2025-12-18DOI: 10.1016/j.disc.2025.114951
Cristina G. Fernandes , Carla N. Lintzmayer , Juan P. Peña , Giovanne Santos , Ana Trujillo-Negrete , Jose Zamora
For a digraph D of order n and an integer , the k-token digraph of D is the digraph whose vertices are all k-subsets of vertices of D and, given two such k-subsets A and B, is an arc in the k-token digraph whenever , , and there is an arc in D. Token digraphs are a generalization of token graphs. In this paper, we study some properties of token digraphs, including strong and unilateral connectivity, kernels, girth, circumference, and Eulerianity. We also extend some known results on the clique and chromatic numbers of k-token graphs, addressing the bidirected clique number and dichromatic number of k-token digraphs. Additionally, we prove that determining whether 2-token digraphs have a kernel is NP-complete.
{"title":"A study on token digraphs","authors":"Cristina G. Fernandes , Carla N. Lintzmayer , Juan P. Peña , Giovanne Santos , Ana Trujillo-Negrete , Jose Zamora","doi":"10.1016/j.disc.2025.114951","DOIUrl":"10.1016/j.disc.2025.114951","url":null,"abstract":"<div><div>For a digraph <em>D</em> of order <em>n</em> and an integer <span><math><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn></math></span>, the <em>k-token digraph</em> of <em>D</em> is the digraph whose vertices are all <em>k</em>-subsets of vertices of <em>D</em> and, given two such <em>k</em>-subsets <em>A</em> and <em>B</em>, <span><math><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo></math></span> is an arc in the <em>k</em>-token digraph whenever <span><math><mo>{</mo><mi>a</mi><mo>}</mo><mo>=</mo><mi>A</mi><mo>∖</mo><mi>B</mi></math></span>, <span><math><mo>{</mo><mi>b</mi><mo>}</mo><mo>=</mo><mi>B</mi><mo>∖</mo><mi>A</mi></math></span>, and there is an arc <span><math><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></math></span> in <em>D</em>. Token digraphs are a generalization of token graphs. In this paper, we study some properties of token digraphs, including strong and unilateral connectivity, kernels, girth, circumference, and Eulerianity. We also extend some known results on the clique and chromatic numbers of <em>k</em>-token graphs, addressing the bidirected clique number and dichromatic number of <em>k</em>-token digraphs. Additionally, we prove that determining whether 2-token digraphs have a kernel is <span>NP</span>-complete.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 5","pages":"Article 114951"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145798256","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.2025.114947
Ilaria Seidel , Nathan Sun
West's stack-sorting map involves a stack which avoids the permutation 21 consecutively. Defant and Zheng extended this to a consecutive-pattern-avoiding stack-sorting map , where the stack avoids a given permutation σ consecutively. We address one of the main conjectures raised by Defant and Zheng in their dynamical approach to . Specifically, we show that the periodic points of are precisely the permutations that consecutively avoid σ and its reverse.
{"title":"Periodic points of consecutive-pattern-avoiding stack-sorting maps","authors":"Ilaria Seidel , Nathan Sun","doi":"10.1016/j.disc.2025.114947","DOIUrl":"10.1016/j.disc.2025.114947","url":null,"abstract":"<div><div>West's stack-sorting map involves a stack which avoids the permutation 21 consecutively. Defant and Zheng extended this to a consecutive-pattern-avoiding stack-sorting map <span><math><mi>S</mi><msub><mrow><mi>C</mi></mrow><mrow><mi>σ</mi></mrow></msub></math></span>, where the stack avoids a given permutation <em>σ</em> consecutively. We address one of the main conjectures raised by Defant and Zheng in their dynamical approach to <span><math><mi>S</mi><msub><mrow><mi>C</mi></mrow><mrow><mi>σ</mi></mrow></msub></math></span>. Specifically, we show that the periodic points of <span><math><mi>S</mi><msub><mrow><mi>C</mi></mrow><mrow><mi>σ</mi></mrow></msub></math></span> are precisely the permutations that consecutively avoid <em>σ</em> and its reverse.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 5","pages":"Article 114947"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145798257","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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