Pub Date : 2024-11-26DOI: 10.1016/j.disc.2024.114334
Jakob Führer , Jozsef Solymosi
Let be a 3-uniform linear hypergraph, i.e. any two edges have at most one vertex common. A special hypergraph, wicket, is formed by three rows and two columns of a point matrix. In this note, we give a new lower bound on the Turán number of wickets using estimates on cap sets. We also show that this problem is closely connected to important questions in additive combinatorics.
{"title":"Caps and wickets","authors":"Jakob Führer , Jozsef Solymosi","doi":"10.1016/j.disc.2024.114334","DOIUrl":"10.1016/j.disc.2024.114334","url":null,"abstract":"<div><div>Let <span><math><msubsup><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></msubsup></math></span> be a 3-uniform linear hypergraph, i.e. any two edges have at most one vertex common. A special hypergraph, <em>wicket</em>, is formed by three rows and two columns of a <span><math><mn>3</mn><mo>×</mo><mn>3</mn></math></span> point matrix. In this note, we give a new lower bound on the Turán number of wickets using estimates on cap sets. We also show that this problem is closely connected to important questions in additive combinatorics.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 3","pages":"Article 114334"},"PeriodicalIF":0.7,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722823","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 : 2024-11-22DOI: 10.1016/j.disc.2024.114331
Bin Chen , Stefanie Gerke , Gregory Gutin , Hui Lei , Heis Parker-Cox , Yacong Zhou
An oriented path P is called anti-directed if every two consecutive arcs of P have opposite orientations. An oriented graph is called k-anti-traceable if every subdigraph induced by k vertices has a hamiltonian anti-directed path. We introduce and study a conjecture, which claims that for every integer there is a least integer such that each k-anti-traceable oriented graph on vertices has a hamiltonian anti-directed path. We determine and show that every k-anti-traceable oriented graph on sufficiently large number n of vertices admits an anti-directed path that contains all but vertices.
如果 P 的每两条连续弧的方向相反，则称为反定向路径 P。如果由 k 个顶点诱导的每个子图都有一条哈密顿反定向路径，则称为 k 反定向图。我们提出并研究了一个猜想，即对于每一个整数 k≥2 都有一个最小整数 f(k)，使得 f(k) 顶点上的每一个 k-anti-traceable 有向图都有一条哈密顿反向路径。我们确定了 f(2),f(3),f(4)，并证明在足够大的 n 个顶点上的每个 k 反跟踪定向图都有一条反定向路径，该路径包含除 o(n) 个顶点之外的所有顶点。
{"title":"On k-anti-traceability of oriented graphs","authors":"Bin Chen , Stefanie Gerke , Gregory Gutin , Hui Lei , Heis Parker-Cox , Yacong Zhou","doi":"10.1016/j.disc.2024.114331","DOIUrl":"10.1016/j.disc.2024.114331","url":null,"abstract":"<div><div>An oriented path <em>P</em> is called anti-directed if every two consecutive arcs of <em>P</em> have opposite orientations. An oriented graph is called <em>k</em>-anti-traceable if every subdigraph induced by <em>k</em> vertices has a hamiltonian anti-directed path. We introduce and study a conjecture, which claims that for every integer <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> there is a least integer <span><math><mi>f</mi><mo>(</mo><mi>k</mi><mo>)</mo></math></span> such that each <em>k</em>-anti-traceable oriented graph on <span><math><mi>f</mi><mo>(</mo><mi>k</mi><mo>)</mo></math></span> vertices has a hamiltonian anti-directed path. We determine <span><math><mi>f</mi><mo>(</mo><mn>2</mn><mo>)</mo><mo>,</mo><mi>f</mi><mo>(</mo><mn>3</mn><mo>)</mo><mo>,</mo><mi>f</mi><mo>(</mo><mn>4</mn><mo>)</mo></math></span> and show that every <em>k</em>-anti-traceable oriented graph on sufficiently large number <em>n</em> of vertices admits an anti-directed path that contains all but <span><math><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> vertices.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 3","pages":"Article 114331"},"PeriodicalIF":0.7,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142701241","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-22DOI: 10.1016/j.disc.2024.114337
Bin Chen
<div><div>In 2006, Alon proposed a problem of characterizing all four-tuples <span><math><mo>(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>d</mi><mo>)</mo></math></span> such that every digraph on <em>n</em> vertices of minimum out-degree at least <em>s</em> contains a subdigraph on <em>m</em> vertices of minimum out-degree at least <em>d</em>. He in particular asked whether there exists an absolute constant <em>c</em> such that every digraph on 2<em>n</em> vertices of minimum out-degree at least <em>s</em> contains a subdigraph on <em>n</em> vertices of minimum out-degree at least <span><math><mfrac><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mi>c</mi></math></span>? Recently, Steiner resolved this case in the negative by showing that for arbitrarily large <em>n</em>, there exists a tournament on 2<em>n</em> vertices of minimum out-degree <span><math><mi>s</mi><mo>=</mo><mi>n</mi><mo>−</mo><mn>1</mn></math></span>, in which the minimum out-degree of every subdigraph on <em>n</em> vertices is at most <span><math><mfrac><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><msub><mrow><mi>log</mi></mrow><mrow><mn>3</mn></mrow></msub><mo></mo><mi>s</mi></math></span>.</div><div>In this paper, we study the above problem and present two new results. The first result is that for arbitrary large <em>n</em> and any integer <span><math><mi>α</mi><mo>≥</mo><mn>2</mn></math></span>, there exists a digraph on <em>αn</em> vertices of minimum out-degree <span><math><mi>s</mi><mo>=</mo><mi>n</mi><mo>−</mo><mn>1</mn></math></span> satisfying that the minimum out-degree of every subdigraph on <em>n</em> vertices is at most <span><math><mfrac><mrow><mi>s</mi></mrow><mrow><mi>α</mi></mrow></mfrac><mo>−</mo><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>α</mi></mrow></mfrac><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><msub><mrow><mi>log</mi></mrow><mrow><mi>α</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mi>s</mi></math></span>. The second result is that for arbitrary large <em>n</em> and any <span><math><mi>r</mi><mo>≥</mo><mn>3</mn></math></span>, there exists a digraph on 2<em>n</em> vertices of girth <em>r</em> and minimum out-degree <em>s</em> satisfying that the minimum out-degree of every subdigraph on <em>n</em> vertices is at most <span><math><mfrac><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><msub><mrow><mi>log</mi></mrow><mrow><mi>r</mi></mrow></msub><mo></mo><mi>s</mi></math></span> if <em>r</em> is odd, and is at most <span><math><mfrac><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>o</mi><mo>(</mo
2006 年，阿隆提出了一个问题：如何描述所有四元组（n,m,s,d），使得 n 个顶点上最小外度至少为 s 的每个图都包含 m 个顶点上最小外度至少为 d 的一个子图？最近，斯坦纳从反面解决了这个问题，他证明了对于任意大的 n，存在一个 2n 个顶点上最小外度为 s=n-1 的锦标赛，其中 n 个顶点上每个子图的最小外度最多为 s2-(12+o(1))log3s.在本文中，我们研究了上述问题，并提出了两个新结果。第一个结果是，对于任意大 n 和任意整数 α≥2，存在一个最小外度为 s=n-1 的 αn 个顶点上的图，它满足 n 个顶点上每个子图的最小外度至多为 sα-(1α+o(1))logα+1s 的要求。第二个结果是，对于任意大 n 和任意 r≥3，存在一个 2n 个顶点上的周长为 r 且最小外度为 s 的图，如果 r 为奇数，则满足 n 个顶点上每个子图的最小外度至多为 s2-(12+o(1))logrs ；如果 r 为偶数，则满足 n 个顶点上每个子图的最小外度至多为 s2-(12+o(1))logr+1s 。
{"title":"New results on a problem of Alon","authors":"Bin Chen","doi":"10.1016/j.disc.2024.114337","DOIUrl":"10.1016/j.disc.2024.114337","url":null,"abstract":"<div><div>In 2006, Alon proposed a problem of characterizing all four-tuples <span><math><mo>(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>d</mi><mo>)</mo></math></span> such that every digraph on <em>n</em> vertices of minimum out-degree at least <em>s</em> contains a subdigraph on <em>m</em> vertices of minimum out-degree at least <em>d</em>. He in particular asked whether there exists an absolute constant <em>c</em> such that every digraph on 2<em>n</em> vertices of minimum out-degree at least <em>s</em> contains a subdigraph on <em>n</em> vertices of minimum out-degree at least <span><math><mfrac><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mi>c</mi></math></span>? Recently, Steiner resolved this case in the negative by showing that for arbitrarily large <em>n</em>, there exists a tournament on 2<em>n</em> vertices of minimum out-degree <span><math><mi>s</mi><mo>=</mo><mi>n</mi><mo>−</mo><mn>1</mn></math></span>, in which the minimum out-degree of every subdigraph on <em>n</em> vertices is at most <span><math><mfrac><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><msub><mrow><mi>log</mi></mrow><mrow><mn>3</mn></mrow></msub><mo></mo><mi>s</mi></math></span>.</div><div>In this paper, we study the above problem and present two new results. The first result is that for arbitrary large <em>n</em> and any integer <span><math><mi>α</mi><mo>≥</mo><mn>2</mn></math></span>, there exists a digraph on <em>αn</em> vertices of minimum out-degree <span><math><mi>s</mi><mo>=</mo><mi>n</mi><mo>−</mo><mn>1</mn></math></span> satisfying that the minimum out-degree of every subdigraph on <em>n</em> vertices is at most <span><math><mfrac><mrow><mi>s</mi></mrow><mrow><mi>α</mi></mrow></mfrac><mo>−</mo><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>α</mi></mrow></mfrac><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><msub><mrow><mi>log</mi></mrow><mrow><mi>α</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mi>s</mi></math></span>. The second result is that for arbitrary large <em>n</em> and any <span><math><mi>r</mi><mo>≥</mo><mn>3</mn></math></span>, there exists a digraph on 2<em>n</em> vertices of girth <em>r</em> and minimum out-degree <em>s</em> satisfying that the minimum out-degree of every subdigraph on <em>n</em> vertices is at most <span><math><mfrac><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><msub><mrow><mi>log</mi></mrow><mrow><mi>r</mi></mrow></msub><mo></mo><mi>s</mi></math></span> if <em>r</em> is odd, and is at most <span><math><mfrac><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>o</mi><mo>(</mo","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 3","pages":"Article 114337"},"PeriodicalIF":0.7,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142701227","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 : 2024-11-19DOI: 10.1016/j.disc.2024.114328
Marc Hellmuth , Bruno J. Schmidt , Guillaume E. Scholz , Sandhya Thekkumpadan Puthiyaveedu
The Djoković-Winkler relation Θ is a binary relation defined on the edge set of a given graph that is based on the distances of certain vertices and which plays a prominent role in graph theory. In this paper, we explore the relatively uncharted “reflexive complement” of Θ, where if and only if or for edges e and f. We establish the relationship between and the set , comprising the distances between the vertices of e and f and shed some light on the intricacies of its transitive closure . Notably, we demonstrate that exhibits multiple equivalence classes only within a restricted subclass of complete multipartite graphs. In addition, we characterize non-trivial relations R that coincide with as those where the graph representation is disconnected, with each connected component being the (join of) Cartesian product of complete graphs. The latter results imply, somewhat surprisingly, that knowledge about the distances between vertices is not required to determine . Moreover, has either exactly one or three equivalence classes.
德约科维奇-温克勒关系Θ是一种定义在给定图边集上的二元关系，它以某些顶点的距离为基础，在图论中发挥着重要作用。在本文中，我们探索了相对未知的Θ的 "反向互补 "Θ‾，其中 (e,f)∈Θ‾ if and only if e=f or (e,f)∉Θ for edges e and f。我们建立了Θ‾与由 e 和 f 的顶点之间的距离组成的集合Δef 之间的关系，并阐明了其传递闭包Θ‾⁎的复杂性。值得注意的是，我们证明了Θ‾⁎仅在完全多方图的有限子类中表现出多个等价类。此外，我们将与Θ‾重合的非三元关系 R 定性为那些图表示是断开的关系，其中每个连通分量都是完整图的（连接）笛卡尔积。后面的结果意味着，在确定Θ‾时不需要了解顶点之间的距离，这有点出人意料。此外，Θ‾⁎ 恰好有一个或三个等价类。
{"title":"The complement of the Djoković-Winkler relation","authors":"Marc Hellmuth , Bruno J. Schmidt , Guillaume E. Scholz , Sandhya Thekkumpadan Puthiyaveedu","doi":"10.1016/j.disc.2024.114328","DOIUrl":"10.1016/j.disc.2024.114328","url":null,"abstract":"<div><div>The Djoković-Winkler relation Θ is a binary relation defined on the edge set of a given graph that is based on the distances of certain vertices and which plays a prominent role in graph theory. In this paper, we explore the relatively uncharted “reflexive complement” <span><math><mover><mrow><mi>Θ</mi></mrow><mo>‾</mo></mover></math></span> of Θ, where <span><math><mo>(</mo><mi>e</mi><mo>,</mo><mi>f</mi><mo>)</mo><mo>∈</mo><mover><mrow><mi>Θ</mi></mrow><mo>‾</mo></mover></math></span> if and only if <span><math><mi>e</mi><mo>=</mo><mi>f</mi></math></span> or <span><math><mo>(</mo><mi>e</mi><mo>,</mo><mi>f</mi><mo>)</mo><mo>∉</mo><mi>Θ</mi></math></span> for edges <em>e</em> and <em>f</em>. We establish the relationship between <span><math><mover><mrow><mi>Θ</mi></mrow><mo>‾</mo></mover></math></span> and the set <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>e</mi><mi>f</mi></mrow></msub></math></span>, comprising the distances between the vertices of <em>e</em> and <em>f</em> and shed some light on the intricacies of its transitive closure <span><math><msup><mrow><mover><mrow><mi>Θ</mi></mrow><mo>‾</mo></mover></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>. Notably, we demonstrate that <span><math><msup><mrow><mover><mrow><mi>Θ</mi></mrow><mo>‾</mo></mover></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> exhibits multiple equivalence classes only within a restricted subclass of complete multipartite graphs. In addition, we characterize non-trivial relations <em>R</em> that coincide with <span><math><mover><mrow><mi>Θ</mi></mrow><mo>‾</mo></mover></math></span> as those where the graph representation is disconnected, with each connected component being the (join of) Cartesian product of complete graphs. The latter results imply, somewhat surprisingly, that knowledge about the distances between vertices is not required to determine <span><math><msup><mrow><mover><mrow><mi>Θ</mi></mrow><mo>‾</mo></mover></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>. Moreover, <span><math><msup><mrow><mover><mrow><mi>Θ</mi></mrow><mo>‾</mo></mover></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> has either exactly one or three equivalence classes.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 3","pages":"Article 114328"},"PeriodicalIF":0.7,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142701173","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-19DOI: 10.1016/j.disc.2024.114327
Wensheng Sun , Yujun Yang , Shou-Jun Xu
A balanced blowup graph of G with respect to a fixed graph H is the graph obtained from G by replacing each vertex with a disjoint copy of H, and connecting each vertex of to each vertex of if there is an edge between and in G. In particular, if H is a complete graph (resp. an empty graph), then is called the clique-blowup (resp. independent-blowup) of G. In (Azimi et al. (2021) [1]), A. Azimi et al. presented combinatorial interpretations of Moore-Penrose inverses of incidence matrices of independent-blowups and clique-blowups of trees, and applied these results to obtain formulas for resistance distances between vertices of the corresponding blowup graphs as well as their Kirchhoff index. In this paper, we extend their results to unbalanced blowup graphs of G via combinatorial and electrical network approaches. In addition, our results contain the main results of (Pan et al. (2021) [17]), (Li et al. (2020) [14]) and (Yan et al. (2023) [26]) as special cases.
相对于固定图 H，G 的平衡炸开图 GH 是将每个顶点 vi∈V(G) 替换为 H 的不相交副本 Hvi，并将 V(Hvi) 的每个顶点连接到 V(Hvj) 的每个顶点（如果在 G 中 vi 和 vj 之间有一条边）而从 G 得到的图。在（Azimi 等人（2021）[1]）一文中，A. Azimi 等人提出了树的独立吹捧和克利克吹捧的入射矩阵的摩尔-彭罗斯倒数的组合解释，并应用这些结果获得了相应吹捧图的顶点之间的阻力距离公式及其基尔霍夫指数。在本文中，我们通过组合和电气网络方法将他们的结果扩展到 G 的不平衡炸开图。此外，我们的结果还包含（Pan 等人 (2021) [17]）、（Li 等人 (2020) [14]）和（Yan 等人 (2023) [26]）作为特例的主要结果。
{"title":"Resistance distance and Kirchhoff index of unbalanced blowups of graphs","authors":"Wensheng Sun , Yujun Yang , Shou-Jun Xu","doi":"10.1016/j.disc.2024.114327","DOIUrl":"10.1016/j.disc.2024.114327","url":null,"abstract":"<div><div>A balanced blowup graph <span><math><msup><mrow><mi>G</mi></mrow><mrow><mi>H</mi></mrow></msup></math></span> of <em>G</em> with respect to a fixed graph <em>H</em> is the graph obtained from <em>G</em> by replacing each vertex <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> with a disjoint copy <span><math><msub><mrow><mi>H</mi></mrow><mrow><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub></math></span> of <em>H</em>, and connecting each vertex of <span><math><mi>V</mi><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub><mo>)</mo></math></span> to each vertex of <span><math><mi>V</mi><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><msub><mrow><mi>v</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub><mo>)</mo></math></span> if there is an edge between <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> in <em>G</em>. In particular, if <em>H</em> is a complete graph (resp. an empty graph), then <span><math><msup><mrow><mi>G</mi></mrow><mrow><mi>H</mi></mrow></msup></math></span> is called the clique-blowup (resp. independent-blowup) of <em>G</em>. In (Azimi et al. (2021) <span><span>[1]</span></span>), A. Azimi et al. presented combinatorial interpretations of Moore-Penrose inverses of incidence matrices of independent-blowups and clique-blowups of trees, and applied these results to obtain formulas for resistance distances between vertices of the corresponding blowup graphs as well as their Kirchhoff index. In this paper, we extend their results to unbalanced blowup graphs of <em>G</em> via combinatorial and electrical network approaches. In addition, our results contain the main results of (Pan et al. (2021) <span><span>[17]</span></span>), (Li et al. (2020) <span><span>[14]</span></span>) and (Yan et al. (2023) <span><span>[26]</span></span>) as special cases.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 3","pages":"Article 114327"},"PeriodicalIF":0.7,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142701240","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 : 2024-11-19DOI: 10.1016/j.disc.2024.114329
Daniel Birmajer , Juan B. Gil , Jordan O. Tirrell , Michael D. Weiner
In this paper, we study pattern avoidance for stabilized-interval-free (SIF) permutations. These permutations are contained in the set of indecomposable permutations and in the set of derangements. We enumerate pattern-avoiding SIF permutations for classical and pairs of patterns of size 3. In particular, for the patterns 123 and 231, we rely on combinatorial arguments and the fixed-point distribution of general permutations avoiding these patterns. We briefly discuss 123-avoiding permutations with two fixed points and offer a conjecture for their enumeration by the distance between their fixed points. For the pattern 231, we also give a direct argument that uses a bijection to ordered forests.
{"title":"Pattern-avoiding stabilized-interval-free permutations","authors":"Daniel Birmajer , Juan B. Gil , Jordan O. Tirrell , Michael D. Weiner","doi":"10.1016/j.disc.2024.114329","DOIUrl":"10.1016/j.disc.2024.114329","url":null,"abstract":"<div><div>In this paper, we study pattern avoidance for stabilized-interval-free (SIF) permutations. These permutations are contained in the set of indecomposable permutations and in the set of derangements. We enumerate pattern-avoiding SIF permutations for classical and pairs of patterns of size 3. In particular, for the patterns 123 and 231, we rely on combinatorial arguments and the fixed-point distribution of general permutations avoiding these patterns. We briefly discuss 123-avoiding permutations with two fixed points and offer a conjecture for their enumeration by the distance between their fixed points. For the pattern 231, we also give a direct argument that uses a bijection to ordered forests.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 3","pages":"Article 114329"},"PeriodicalIF":0.7,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142701172","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 : 2024-11-16DOI: 10.1016/j.disc.2024.114326
Thiago Assis, Gabriel Coutinho, Emanuel Juliano
The Grundy number of a graph is the minimum number of colors needed to properly color the graph using the first-fit greedy algorithm regardless of the initial vertex ordering. Computing the Grundy number of a graph is an NP-Hard problem. There is a characterization in terms of induced subgraphs: a graph has a Grundy number at least k if and only if it contains a k-atom. In this paper, using properties of the matching polynomial, we determine the smallest possible largest eigenvalue of a k-atom. With this result, we present an upper bound for the Grundy number of a graph in terms of the largest eigenvalue of its adjacency matrix. We also present another upper bound using the largest eigenvalue and the size of the graph. Our bounds are asymptotically tight for some infinite families of graphs and provide improvements on the known bounds for the Grundy number of sparse random graphs.
图形的格兰迪数是指在不考虑初始顶点排序的情况下，使用第一拟合贪婪算法为图形正确着色所需的最少颜色数。计算图形的格兰迪数是一个 NP-Hard（近乎困难）问题。从诱导子图的角度来看，有这样一个特征：当且仅当一个图包含一个 k 原子时，该图的格兰迪数至少为 k。在本文中，我们利用匹配多项式的特性，确定了 k 原子的最小最大特征值。有了这一结果，我们根据图的邻接矩阵的最大特征值提出了图的格兰迪数上限。我们还利用最大特征值和图的大小提出了另一个上限。对于某些无限图族，我们的上界是渐近紧密的，并且改进了稀疏随机图的格兰迪数的已知上界。
{"title":"Spectral upper bounds for the Grundy number of a graph","authors":"Thiago Assis, Gabriel Coutinho, Emanuel Juliano","doi":"10.1016/j.disc.2024.114326","DOIUrl":"10.1016/j.disc.2024.114326","url":null,"abstract":"<div><div>The Grundy number of a graph is the minimum number of colors needed to properly color the graph using the first-fit greedy algorithm regardless of the initial vertex ordering. Computing the Grundy number of a graph is an NP-Hard problem. There is a characterization in terms of induced subgraphs: a graph has a Grundy number at least k if and only if it contains a <em>k</em>-atom. In this paper, using properties of the matching polynomial, we determine the smallest possible largest eigenvalue of a <em>k</em>-atom. With this result, we present an upper bound for the Grundy number of a graph in terms of the largest eigenvalue of its adjacency matrix. We also present another upper bound using the largest eigenvalue and the size of the graph. Our bounds are asymptotically tight for some infinite families of graphs and provide improvements on the known bounds for the Grundy number of sparse random graphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 3","pages":"Article 114326"},"PeriodicalIF":0.7,"publicationDate":"2024-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142661897","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 : 2024-11-16DOI: 10.1016/j.disc.2024.114330
Daniel R. Hawtin
A t-fold packing of a projective space is a collection of line-spreads such that each line of occurs in precisely t spreads in . A t-fold packing is transitive if a subgroup of preserves and acts transitively on . We give a construction for a transitive -fold packing of , where , for any odd positive integers n and k, such that . This generalises a construction of Baker from 1976 for the case .
如果 PΓLn+1(q)的一个子群保存并在 P 上起传递作用，那么一个 t 折叠包装 P 就是传递性的。我们给出了一个 PGn(q)的传递性 (q-1)-fold 包装的构造，其中 q=2k, 对于任何奇数正整数 n 和 k，使得 n⩾3 。这概括了贝克 1976 年针对 q=2 情况的构造。
{"title":"Transitive (q − 1)-fold packings of PGn(q)","authors":"Daniel R. Hawtin","doi":"10.1016/j.disc.2024.114330","DOIUrl":"10.1016/j.disc.2024.114330","url":null,"abstract":"<div><div>A <em>t-fold packing</em> of a projective space <span><math><msub><mrow><mi>PG</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span> is a collection <span><math><mi>P</mi></math></span> of line-spreads such that each line of <span><math><msub><mrow><mi>PG</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span> occurs in precisely <em>t</em> spreads in <span><math><mi>P</mi></math></span>. A <em>t</em>-fold packing <span><math><mi>P</mi></math></span> is <em>transitive</em> if a subgroup of <span><math><msub><mrow><mi>P</mi><mi>Γ</mi><mi>L</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span> preserves and acts transitively on <span><math><mi>P</mi></math></span>. We give a construction for a transitive <span><math><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-fold packing of <span><math><msub><mrow><mi>PG</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span>, where <span><math><mi>q</mi><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msup></math></span>, for any odd positive integers <em>n</em> and <em>k</em>, such that <span><math><mi>n</mi><mo>⩾</mo><mn>3</mn></math></span>. This generalises a construction of Baker from 1976 for the case <span><math><mi>q</mi><mo>=</mo><mn>2</mn></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 3","pages":"Article 114330"},"PeriodicalIF":0.7,"publicationDate":"2024-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142661898","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 : 2024-11-14DOI: 10.1016/j.disc.2024.114319
Cristina Ballantine , Brooke Feigon
The work of Andrews and Merca on the truncated Euler's pentagonal number theorem led to a resurgence in research on truncated theta series identities. In particular, Yee proved a truncated version of the Jacobi Triple Product (JTP) identity. Recently, Merca conjectured a stronger form of the truncated JTP identity. In this article we prove the first three cases of the conjecture and several related truncated identities. We prove combinatorially an identity related to the JTP identity which in particular cases reduces to identities conjectured by Merca and proved analytically by Krattenthaler, Merca and Radu. Moreover, we introduce a new combinatorial interpretation for the number of distinct 5-regular partitions of n.
{"title":"Truncated theta series related to the Jacobi Triple Product identity","authors":"Cristina Ballantine , Brooke Feigon","doi":"10.1016/j.disc.2024.114319","DOIUrl":"10.1016/j.disc.2024.114319","url":null,"abstract":"<div><div>The work of Andrews and Merca on the truncated Euler's pentagonal number theorem led to a resurgence in research on truncated theta series identities. In particular, Yee proved a truncated version of the Jacobi Triple Product (JTP) identity. Recently, Merca conjectured a stronger form of the truncated JTP identity. In this article we prove the first three cases of the conjecture and several related truncated identities. We prove combinatorially an identity related to the JTP identity which in particular cases reduces to identities conjectured by Merca and proved analytically by Krattenthaler, Merca and Radu. Moreover, we introduce a new combinatorial interpretation for the number of distinct 5-regular partitions of <em>n</em>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114319"},"PeriodicalIF":0.7,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142655715","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 : 2024-11-13DOI: 10.1016/j.disc.2024.114322
Stefan Mitrović, Tanja Stojadinović
We introduce two classes of graphs - suns and dumbbells, both with few variations and explore their chromatic symmetric function and its e-positivity. We also give many connections of these two classes with other classes of connected graphs.
我们介绍了两类图--太阳图和哑铃图，它们都有很少的变化，并探讨了它们的色度对称函数及其 e 正性。我们还给出了这两类图与其他连通图的许多联系。
{"title":"The e−positivity of some new classes of graphs","authors":"Stefan Mitrović, Tanja Stojadinović","doi":"10.1016/j.disc.2024.114322","DOIUrl":"10.1016/j.disc.2024.114322","url":null,"abstract":"<div><div>We introduce two classes of graphs - suns and dumbbells, both with few variations and explore their chromatic symmetric function and its <em>e</em>-positivity. We also give many connections of these two classes with other classes of connected graphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 3","pages":"Article 114322"},"PeriodicalIF":0.7,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142661895","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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