Graphs and Combinatorics最新文献_第3页

Cliques of Orders Three and Four in the Paley-Type Graphs 帕利型图中的三阶和四阶小群

IF 0.7 4区数学

Graphs and Combinatorics Pub Date : 2024-06-24 DOI: 10.1007/s00373-024-02809-1

Anwita Bhowmik, Rupam Barman

{"title":"Cliques of Orders Three and Four in the Paley-Type Graphs","authors":"Anwita Bhowmik, Rupam Barman","doi":"10.1007/s00373-024-02809-1","DOIUrl":"https://doi.org/10.1007/s00373-024-02809-1","url":null,"abstract":"Let (n=2^s p_{1}^{alpha _{1}}cdots p_{k}^{alpha _{k}}), where (s=0) or 1, (alpha _ige 1), and the distinct primes (p_i) satisfy (p_iequiv 1pmod {4}) for all (i=1, ldots , k). Let (mathbb {Z}_n^*) denote the group of units in the commutative ring (mathbb {Z}_n). In a recent paper, we defined the Paley-type graph (G_n) of order n as the graph whose vertex set is (mathbb {Z}_n) and xy is an edge if (x-yequiv a^2pmod n) for some (ain mathbb {Z}_n^*). Computing the number of cliques of a particular order in a Paley graph or its generalizations has been of considerable interest. In our recent paper, for primes (pequiv 1pmod 4) and (alpha ge 1), by evaluating certain character sums, we found the number of cliques of order 3 in (G_{p^alpha }) and expressed the number of cliques of order 4 in (G_{p^alpha }) in terms of Jacobi sums. In this article we give combinatorial proofs and find the number of cliques of orders 3 and 4 in (G_n) for all n for which the graph is defined.","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":"126 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141505150","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Partitions of Vertices and Facets in Trees and Stacked Simplicial Complexes 树和堆叠简复中顶点和面的分区

IF 0.7 4区数学

Graphs and Combinatorics Pub Date : 2024-06-18 DOI: 10.1007/s00373-024-02804-6

Gunnar Fløystad

引用次数: 0

Pivot Gray Codes for the Spanning Trees of a Graph ft. the Fan 图的生成树的枢轴灰色代码英尺扇形

IF 0.7 4区数学

Graphs and Combinatorics Pub Date : 2024-06-11 DOI: 10.1007/s00373-024-02808-2

Ben Cameron, Aaron Grubb, Joe Sawada

引用次数: 0

Injective Chromatic Index of $$K_4$$ -Minor Free Graphs $$K_4$$ 小自由图的注入色度指数

IF 0.7 4区数学

Graphs and Combinatorics Pub Date : 2024-06-04 DOI: 10.1007/s00373-024-02807-3

Jian-Bo Lv, Jiacong Fu, Jianxi Li

引用次数: 0

A Dirac-Type Theorem for Uniform Hypergraphs 均匀超图的狄拉克定理

IF 0.7 4区数学

Graphs and Combinatorics Pub Date : 2024-06-01 DOI: 10.1007/s00373-024-02802-8

Yue Ma, Xinmin Hou, Jun Gao

{"title":"A Dirac-Type Theorem for Uniform Hypergraphs","authors":"Yue Ma, Xinmin Hou, Jun Gao","doi":"10.1007/s00373-024-02802-8","DOIUrl":"https://doi.org/10.1007/s00373-024-02802-8","url":null,"abstract":"Dirac (Proc Lond Math Soc (3) 2:69–81, 1952) proved that every connected graph of order (n>2k+1) with minimum degree more than k contains a path of length at least (2k+1). In this article, we give a hypergraph extension of Dirac’s theorem: Given positive integers n, k and r, let H be a connected n-vertex r-graph with no Berge path of length (2k+1). (1) If (k> rge 4) and (n>2k+1), then (delta _1(H)le left( {begin{array}{c}k r-1end{array}}right) ). Furthermore, there exist hypergraphs (S'_r(n,k), S_r(n,k)) and (S(sK_{k+1}^{(r)},1)) such that the equality holds if and only if (S'_r(n,k)subseteq Hsubseteq S_r(n,k)) or (Hcong S(sK_{k+1}^{(r)},1)); (2) If (kge rge 2) and (n>2k(r-1)), then (delta _1(H)le left( {begin{array}{c}k r-1end{array}}right) ). As an application of (1), we give a better lower bound of the minimum degree than the ones in the Dirac-type results for Berge Hamiltonian cycle given by Bermond et al. (Hypergraphes Hamiltoniens. In: Problémes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976). Colloq. Internat. CNRS, vol. 260, pp. 39–43. CNRS, Paris, 1976) or Clemens et al. (Electron Notes Discrete Math 54:181–186, 2016), respectively.","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":"48 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141194204","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

The Cordiality Game and the Game Cordiality Number 亲切游戏和游戏亲切号码

IF 0.7 4区数学

Graphs and Combinatorics Pub Date : 2024-05-30 DOI: 10.1007/s00373-024-02798-1

Elliot Krop, Aryan Mittal, Michael C. Wigal

引用次数: 0

Irreducible Pairings and Indecomposable Tournaments 不可还原配对和不可分解锦标赛

IF 0.7 4区数学

Graphs and Combinatorics Pub Date : 2024-05-30 DOI: 10.1007/s00373-024-02803-7

Houmem Belkhechine, Cherifa Ben Salha, Rim Romdhane

{"title":"Irreducible Pairings and Indecomposable Tournaments","authors":"Houmem Belkhechine, Cherifa Ben Salha, Rim Romdhane","doi":"10.1007/s00373-024-02803-7","DOIUrl":"https://doi.org/10.1007/s00373-024-02803-7","url":null,"abstract":"We only consider finite structures. With every totally ordered set V and a subset P of (left( {begin{array}{c}V 2end{array}}right) ), we associate the underlying tournament (textrm{Inv}({underline{V}}, P)) obtained from the transitive tournament ({underline{V}}:=(V, {(x,y) in V times V: x < y })) by reversing P, i.e., by reversing the arcs (x, y) such that ({x,y} in P). The subset P is a pairing (of (cup P)) if (|cup P| = 2|P|), a quasi-pairing (of (cup P)) if (|cup P| = 2|P|-1); it is irreducible if no nontrivial interval of (cup P) is a union of connected components of the graph ((cup P, P)). In this paper, we consider pairings and quasi-pairings in relation to tournaments. We establish close relationships between irreducibility of pairings (or quasi-pairings) and indecomposability of their underlying tournaments under modular decomposition. For example, given a pairing P of a totally ordered set V of size at least 6, the pairing P is irreducible if and only if the tournament (textrm{Inv}({underline{V}}, P)) is indecomposable. This is a consequence of a more general result characterizing indecomposable tournaments obtained from transitive tournaments by reversing pairings. We obtain analogous results in the case of quasi-pairings.","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":"57 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141194203","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A Characterization of Graphs with Semitotal Domination Number One-Third Their Order 具有三分之一阶半总支配数的图的特征

IF 0.7 4区数学

Graphs and Combinatorics Pub Date : 2024-05-30 DOI: 10.1007/s00373-024-02800-w

Jie Chen, Cai-Xia Wang, Yi-Ping Liang, Shou-Jun Xu

引用次数: 0

Transitive Subtournaments of k-th Power Paley Digraphs and Improved Lower Bounds for Ramsey Numbers k-th Power Paley Digraph 的跨域子域和改进的拉姆齐数下限

IF 0.7 4区数学

Graphs and Combinatorics Pub Date : 2024-05-21 DOI: 10.1007/s00373-024-02792-7

Dermot McCarthy, Mason Springfield

{"title":"Transitive Subtournaments of k-th Power Paley Digraphs and Improved Lower Bounds for Ramsey Numbers","authors":"Dermot McCarthy, Mason Springfield","doi":"10.1007/s00373-024-02792-7","DOIUrl":"https://doi.org/10.1007/s00373-024-02792-7","url":null,"abstract":"Let (k ge 2) be an even integer. Let q be a prime power such that (q equiv k+1 (text {mod},,2k)). We define the k-th power Paley digraph of order q, (G_k(q)), as the graph with vertex set (mathbb {F}_q) where (a rightarrow b) is an edge if and only if (b-a) is a k-th power residue. This generalizes the ((k=2)) Paley Tournament. We provide a formula, in terms of finite field hypergeometric functions, for the number of transitive subtournaments of order four contained in (G_k(q)), (mathcal {K}_4(G_k(q))), which holds for all k. We also provide a formula, in terms of Jacobi sums, for the number of transitive subtournaments of order three contained in (G_k(q)), (mathcal {K}_3(G_k(q))). In both cases, we give explicit determinations of these formulae for small k. We show that zero values of (mathcal {K}_4(G_k(q))) (resp. (mathcal {K}_3(G_k(q)))) yield lower bounds for the multicolor directed Ramsey numbers (R_{frac{k}{2}}(4)=R(4,4,ldots ,4)) (resp. (R_{frac{k}{2}}(3))). We state explicitly these lower bounds for (kle 10) and compare to known bounds, showing improvement for (R_2(4)) and (R_3(3)). Combining with known multiplicative relations we give improved lower bounds for (R_{t}(4)), for all (tge 2), and for (R_{t}(3)), for all (t ge 3).","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":"37 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141150800","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

The Generalized Terwilliger Algebra of the Hypercube 超立方体的广义特尔维利格代数

IF 0.7 4区数学

Graphs and Combinatorics Pub Date : 2024-05-20 DOI: 10.1007/s00373-024-02801-9

Nathan Nicholson

引用次数: 0