Xiaoyu Li , Rong-Xia Hao , Rong Luo , Cun-Quan Zhang
{"title":"Non-separating cycles and 5-cycle double covers","authors":"Xiaoyu Li , Rong-Xia Hao , Rong Luo , 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, Rong-Xia Hao, Zhen He, 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}
{"title":"Decomposition family and spectral extremal problems on non-bipartite graphs","authors":"Longfei Fang , Michael Tait , Mingqing Zhai","doi":"10.1016/j.disc.2025.114527","DOIUrl":"10.1016/j.disc.2025.114527","url":null,"abstract":"<div><div>Given a graph family <span><math><mi>H</mi></math></span> with <span><math><msub><mrow><mi>min</mi></mrow><mrow><mi>H</mi><mo>∈</mo><mi>H</mi></mrow></msub><mo></mo><mi>χ</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>=</mo><mi>r</mi><mo>+</mo><mn>1</mn><mo>≥</mo><mn>3</mn></math></span>. Let <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> and <span><math><mrow><mi>spex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> be the maximum number of edges and the maximum spectral radius of the adjacency matrix over all <em>n</em>-vertex <span><math><mi>H</mi></math></span>-free graphs, respectively. Denote by <span><math><mrow><mi>EX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> (resp. <span><math><mrow><mi>SPEX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span>) the set of extremal graphs with respect to <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> (resp. <span><math><mrow><mi>spex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span>). A fundamental problem in extremal spectral graph theory asks which graph <em>H</em> satisfies <span><math><mrow><mi>SPEX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo><mo>⊆</mo><mrow><mi>EX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span>.</div><div>Wang et al. (2023) <span><span>[43]</span></span> proved that <span><math><mrow><mi>SPEX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo><mo>⊆</mo><mrow><mi>EX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> for <em>n</em> sufficiently large and any finite graph <em>H</em> with <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo><mo>=</mo><mi>e</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>r</mi></mrow></msub><mo>)</mo><mo>+</mo><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span>. In this paper, we use decomposition family defined by Simonovits to give a characterization of which graph families <span><math><mi>H</mi></math></span> satisfy <span><math><mi>e</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>r</mi></mrow></msub><mo>)</mo><mo>≤</mo><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo><mo><</mo><mi>e</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>r</mi></mrow></msub><mo>)</mo><mo>+</mo><mo>⌊</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn><mi>r</mi></mrow></mfrac><mo>⌋</mo></math></span>. Using this result, we show that <span><math><mrow><mi>SPEX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo><mo>⊆</mo><mrow><mi>EX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> for <em>n</em> sufficiently large and any finite family <span><math><mi>H</mi></math></span> with <span><math><mi>e</mi><mo>(</mo><ms","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 10","pages":"Article 114527"},"PeriodicalIF":0.7,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143808790","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}
Jinzhuan Cai , Jin Guo , Alexander L. Gavrilyuk , Ilia Ponomarenko
{"title":"Cartesian products of graphs and their coherent configurations","authors":"Jinzhuan Cai , Jin Guo , Alexander L. Gavrilyuk , Ilia Ponomarenko","doi":"10.1016/j.disc.2025.114526","DOIUrl":"10.1016/j.disc.2025.114526","url":null,"abstract":"<div><div>The coherent configuration <span><math><mi>WL</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> of a graph <em>X</em> is the smallest coherent configuration on the vertices of <em>X</em> that contains the edge set of <em>X</em> as a relation. The aim of the paper is to study <span><math><mi>WL</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> when <em>X</em> is a Cartesian product of graphs. The example of a Hamming graph shows that, in general, <span><math><mi>WL</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> does not coincide with the tensor product of the coherent configurations of the factors. We prove that if <em>X</em> is “closed” with respect to the 6-dimensional Weisfeiler-Leman algorithm, then <span><math><mi>WL</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is the tensor product of the coherent configurations of certain graphs related to the prime decomposition of <em>X</em>. This condition is trivially satisfied for almost all graphs. In addition, we prove that the property of a graph “to be decomposable into a Cartesian product of <em>k</em> connected prime graphs” for some <span><math><mi>k</mi><mo>≥</mo><mn>1</mn></math></span> is recognized by the <em>m</em>-dimensional Weisfeiler-Leman algorithm for all <span><math><mi>m</mi><mo>≥</mo><mn>6</mn></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114526"},"PeriodicalIF":0.7,"publicationDate":"2025-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143807409","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 dual codes of negacyclic BCH codes of length qm+12","authors":"Yuqing Fu , Hongwei Liu","doi":"10.1016/j.disc.2025.114525","DOIUrl":"10.1016/j.disc.2025.114525","url":null,"abstract":"<div><div>Negacyclic BCH codes form an important subclass of negacyclic codes and can produce optimal linear codes in many cases. The question of whether the dual code of a negacyclic BCH code is a negacyclic BCH code is, in general, very hard to answer. To investigate further the properties of the dual codes of negacyclic BCH codes, the concept of negacyclic dually-BCH codes is proposed in this paper and then the dual codes of narrow-sense negacyclic BCH codes of length <span><math><mfrac><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> over the finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> are studied, where <span><math><mi>q</mi><mo>≡</mo><mn>3</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>4</mn><mo>)</mo></math></span>. Some lower bounds on the minimum distances of the dual codes are established, which are very close to the true minimum distances of the dual codes in many cases. Sufficient and necessary conditions in terms of designed distances are presented to ensure that narrow-sense negacyclic BCH codes of length <span><math><mfrac><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> are negacyclic dually-BCH codes.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 9","pages":"Article 114525"},"PeriodicalIF":0.7,"publicationDate":"2025-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143799853","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":"Thresholds for pebbling on grids","authors":"Neal Bushaw , Nathan Kettle","doi":"10.1016/j.disc.2025.114519","DOIUrl":"10.1016/j.disc.2025.114519","url":null,"abstract":"<div><div>Given a connected graph <em>G</em> and a configuration of <em>t</em> pebbles on the vertices of G, a <em>q</em>-pebbling step consists of removing <em>q</em> pebbles from a vertex, and adding a single pebble to one of its neighbors. Given a vector <span><math><mi>q</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>q</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>q</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo></math></span>, <em>q</em>-pebbling consists of allowing <span><math><msub><mrow><mi>q</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>-pebbling in coordinate <em>i</em>. A distribution of pebbles is called solvable if it is possible to transfer at least one pebble to any specified vertex of <em>G</em> via a finite sequence of pebbling steps.</div><div>In this paper, we determine the weak threshold for <strong>q</strong>-pebbling on the sequence of grids <span><math><msup><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow><mrow><mi>d</mi></mrow></msup></math></span> for fixed <em>d</em> and <strong>q</strong>, as <span><math><mi>n</mi><mo>→</mo><mo>∞</mo></math></span>. Further, we determine the strong threshold for <em>q</em>-pebbling on the sequence of paths of increasing length. A fundamental tool in these proofs is a new notion of ‘centralness’ and a sufficient condition for solvability based on the well used pebbling weight functions; we believe this to be the first result of its kind, and may be of independent interest.</div><div>These theorems improve recent results of Czygrinow and Hurlbert, and Godbole, Jablonski, Salzman, and Wierman. They are the generalizations to the random setting of much earlier results of Chung.</div><div>In addition, we give a short counterexample showing that the threshold version of a well known conjecture of Graham does not hold. This uses a result for hypercubes due to Czygrinow and Wagner.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 10","pages":"Article 114519"},"PeriodicalIF":0.7,"publicationDate":"2025-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143799497","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":"Brualdi–Hoffman–Turán problem of the gem","authors":"Fan Chen , Xiying Yuan","doi":"10.1016/j.disc.2025.114528","DOIUrl":"10.1016/j.disc.2025.114528","url":null,"abstract":"<div><div>A graph is said to be <em>F</em>-free if it does not contain <em>F</em> as a subgraph. Brualdi–Hoffman– Turán type problem seeks to determine the maximum spectral radius of an <em>F</em>-free graph with given size. The gem consists of a path on 4 vertices, along with an additional vertex that is adjacent to every vertex of the path. Concerning Brualdi–Hoffman–Turán type problem of the gem, when the size is odd, Zhang and Wang (2024) <span><span>[20]</span></span> and Yu et al. (2025) <span><span>[18]</span></span> solved it. In this paper, we completely solve the Brualdi–Hoffman–Turán type problem of the gem.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 10","pages":"Article 114528"},"PeriodicalIF":0.7,"publicationDate":"2025-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143799496","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}
Ervin Győri , Ryan R. Martin , Addisu Paulos , Casey Tompkins , Kitti Varga
{"title":"On the rainbow planar Turán number of paths","authors":"Ervin Győri , Ryan R. Martin , Addisu Paulos , Casey Tompkins , Kitti Varga","doi":"10.1016/j.disc.2025.114523","DOIUrl":"10.1016/j.disc.2025.114523","url":null,"abstract":"<div><div>An edge-colored graph is said to contain a rainbow-<em>F</em> if it contains <em>F</em> as a subgraph and every edge of <em>F</em> is a distinct color. The problem of maximizing the number of edges among <em>n</em>-vertex properly edge-colored graphs not containing a rainbow-<em>F</em>, known as the rainbow Turán problem, was initiated by Keevash, Mubayi, Sudakov, and Verstraëte. We investigate a variation of this problem with the additional restriction that the graph is planar and we denote the corresponding extremal number by <span><math><msubsup><mrow><mi>ex</mi></mrow><mrow><mi>P</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span>. In particular, we determine <span><math><msubsup><mrow><mi>ex</mi></mrow><mrow><mi>P</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span> denotes the 5-vertex path.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 10","pages":"Article 114523"},"PeriodicalIF":0.7,"publicationDate":"2025-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143799495","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the combinatorial structure and algebraic characterizations of distance-regular digraphs","authors":"Giusy Monzillo , Safet Penjić","doi":"10.1016/j.disc.2025.114512","DOIUrl":"10.1016/j.disc.2025.114512","url":null,"abstract":"<div><div>Let <span><math><mi>Γ</mi><mo>=</mo><mi>Γ</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> denote a simple strongly connected digraph with vertex set <em>X</em>, diameter <em>D</em>, and let <span><math><mo>{</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mi>A</mi><mo>:</mo><mo>=</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>D</mi></mrow></msub><mo>}</mo></math></span> denote the set of distance-<em>i</em> matrices of Γ. Let <span><math><msubsup><mrow><mo>{</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>D</mi></mrow></msubsup></math></span> denote a partition of <span><math><mi>X</mi><mo>×</mo><mi>X</mi></math></span>, where <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mo>{</mo><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>∈</mo><mi>X</mi><mo>×</mo><mi>X</mi><mo>|</mo><msub><mrow><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>=</mo><mn>1</mn><mo>}</mo></math></span> <span><math><mo>(</mo><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>D</mi><mo>)</mo></math></span>. In the literature, such a digraph Γ is said to be <em>distance-regular</em> if <span><math><mo>(</mo><mi>X</mi><mo>,</mo><msubsup><mrow><mo>{</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>D</mi></mrow></msubsup><mo>)</mo></math></span> is a commutative association scheme. In this paper, we provide a combinatorial definition of a distance-regular digraph in terms of equitable partitions. From this definition, we rediscover all well-known algebraic characterizations of such digraphs, including the above one. We also give several new characterizations, and one of them is the spectral excess theorem for distance-regular digraphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114512"},"PeriodicalIF":0.7,"publicationDate":"2025-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143786053","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":"Synchronicity of descent and excedance enumerators in the alternating subgroup","authors":"Umesh Shankar","doi":"10.1016/j.disc.2025.114521","DOIUrl":"10.1016/j.disc.2025.114521","url":null,"abstract":"<div><div>Generalising the work of Dey <span><span>[2]</span></span>, we define the notion of ultra-synchronicity of sequences of real numbers. Let <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub></math></span> be the number of even permutations of <span><math><mo>[</mo><mi>n</mi><mo>]</mo></math></span> with <em>k</em> descents, odd permutations with <em>k</em> descents, even permutations with <em>k</em> excedances and odd permutations with <em>k</em> excedances, respectively. We show that the four sequences are ultra-synchronised for all <span><math><mi>n</mi><mo>≥</mo><mn>5</mn></math></span>. This proves a strengthening of two conjectures of Dey <span><span>[2]</span></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114521"},"PeriodicalIF":0.7,"publicationDate":"2025-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143786052","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}
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
