Pub Date : 2024-06-24DOI: 10.1007/s00373-024-02809-1
Anwita Bhowmik, Rupam Barman
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.
{"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":"<p>Let <span>(n=2^s p_{1}^{alpha _{1}}cdots p_{k}^{alpha _{k}})</span>, where <span>(s=0)</span> or 1, <span>(alpha _ige 1)</span>, and the distinct primes <span>(p_i)</span> satisfy <span>(p_iequiv 1pmod {4})</span> for all <span>(i=1, ldots , k)</span>. Let <span>(mathbb {Z}_n^*)</span> denote the group of units in the commutative ring <span>(mathbb {Z}_n)</span>. In a recent paper, we defined the Paley-type graph <span>(G_n)</span> of order <i>n</i> as the graph whose vertex set is <span>(mathbb {Z}_n)</span> and <i>xy</i> is an edge if <span>(x-yequiv a^2pmod n)</span> for some <span>(ain mathbb {Z}_n^*)</span>. 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 <span>(pequiv 1pmod 4)</span> and <span>(alpha ge 1)</span>, by evaluating certain character sums, we found the number of cliques of order 3 in <span>(G_{p^alpha })</span> and expressed the number of cliques of order 4 in <span>(G_{p^alpha })</span> in terms of Jacobi sums. In this article we give combinatorial proofs and find the number of cliques of orders 3 and 4 in <span>(G_n)</span> for all <i>n</i> for which the graph is defined.</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":null,"pages":null},"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}
Pub Date : 2024-06-18DOI: 10.1007/s00373-024-02804-6
Gunnar Fløystad
For stacked simplicial complexes, (special subclasses of such are: trees, triangulations of polygons, stacked polytopes with their triangulations), we give an explicit bijection between partitions of facets (for trees: edges), and partitions of vertices into independent sets. More generally, we give bijections between facet partitions whose parts have minimal distance (ge s) and vertex partitions whose parts have minimal distance (ge s+1).
{"title":"Partitions of Vertices and Facets in Trees and Stacked Simplicial Complexes","authors":"Gunnar Fløystad","doi":"10.1007/s00373-024-02804-6","DOIUrl":"https://doi.org/10.1007/s00373-024-02804-6","url":null,"abstract":"<p>For stacked simplicial complexes, (special subclasses of such are: trees, triangulations of polygons, stacked polytopes with their triangulations), we give an explicit bijection between partitions of facets (for trees: edges), and partitions of vertices into independent sets. More generally, we give bijections between facet partitions whose parts have minimal distance <span>(ge s)</span> and vertex partitions whose parts have minimal distance <span>(ge s+1)</span>.</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141529213","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}
Pub Date : 2024-06-11DOI: 10.1007/s00373-024-02808-2
Ben Cameron, Aaron Grubb, Joe Sawada
We consider the problem of listing all spanning trees of a graph G such that successive trees differ by pivoting a single edge around a vertex. Such a listing is called a “pivot Gray code”, and it has more stringent conditions than known “revolving-door” Gray codes for spanning trees. Most revolving-door algorithms employ a standard edge-deletion/edge-contraction recursive approach which we demonstrate presents natural challenges when requiring the “pivot” property. Our main result is the discovery of a greedy strategy to list the spanning trees of the fan graph in a pivot Gray code order. It is the first greedy algorithm for exhaustively generating spanning trees using such a minimal change operation. The resulting listing is then studied to find a recursive algorithm that produces the same listing in O(1)-amortized time using O(n) space. Additionally, we present O(n)-time algorithms for ranking and unranking the spanning trees for our listing. Finally, we discuss how our listing can be applied to find a pivot Gray code for the wheel graph.
{"title":"Pivot Gray Codes for the Spanning Trees of a Graph ft. the Fan","authors":"Ben Cameron, Aaron Grubb, Joe Sawada","doi":"10.1007/s00373-024-02808-2","DOIUrl":"https://doi.org/10.1007/s00373-024-02808-2","url":null,"abstract":"<p>We consider the problem of listing all spanning trees of a graph <i>G</i> such that successive trees differ by pivoting a single edge around a vertex. Such a listing is called a “pivot Gray code”, and it has more stringent conditions than known “revolving-door” Gray codes for spanning trees. Most revolving-door algorithms employ a standard edge-deletion/edge-contraction recursive approach which we demonstrate presents natural challenges when requiring the “pivot” property. Our main result is the discovery of a greedy strategy to list the spanning trees of the fan graph in a pivot Gray code order. It is the first greedy algorithm for exhaustively generating spanning trees using such a minimal change operation. The resulting listing is then studied to find a recursive algorithm that produces the same listing in <i>O</i>(1)-amortized time using <i>O</i>(<i>n</i>) space. Additionally, we present <i>O</i>(<i>n</i>)-time algorithms for ranking and unranking the spanning trees for our listing. Finally, we discuss how our listing can be applied to find a pivot Gray code for the wheel graph.</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141505151","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}
Pub Date : 2024-06-04DOI: 10.1007/s00373-024-02807-3
Jian-Bo Lv, Jiacong Fu, Jianxi Li
An edge-coloring of a graph G is injective if for any two distinct edges (e_1) and (e_2), the colors of (e_1) and (e_2) are distinct if they are at distance 2 in G or in a common triangle. The injective chromatic index of G, (chi ^prime _{inj}(G)), is the minimum number of colors needed for an injective edge-coloring of G. In this note, we show that every (K_4)-minor free graph G with maximum degree (Delta (G)ge 3) satisfies (chi ^prime _{inj}(G)le 2Delta (G)+1).
如果对于任意两条不同的边(e_1)和(e_2)来说,如果它们在 G 中的距离是 2 或者在一个共同的三角形中,那么它们的颜色就是不同的,那么图 G 的边着色就是可注入的。G 的注入色度指数((chi ^prime _{inj}(G)))是 G 的注入边着色所需的最少颜色数。在本说明中,我们证明了每个具有最大度的(Delta (G)ge 3) 的(K_4)-minor free graph G 都满足(chi ^prime _{inj}(G)le 2Delta (G)+1)。
{"title":"Injective Chromatic Index of $$K_4$$ -Minor Free Graphs","authors":"Jian-Bo Lv, Jiacong Fu, Jianxi Li","doi":"10.1007/s00373-024-02807-3","DOIUrl":"https://doi.org/10.1007/s00373-024-02807-3","url":null,"abstract":"<p>An edge-coloring of a graph <i>G</i> is <i>injective</i> if for any two distinct edges <span>(e_1)</span> and <span>(e_2)</span>, the colors of <span>(e_1)</span> and <span>(e_2)</span> are distinct if they are at distance 2 in <i>G</i> or in a common triangle. The injective chromatic index of <i>G</i>, <span>(chi ^prime _{inj}(G))</span>, is the minimum number of colors needed for an injective edge-coloring of <i>G</i>. In this note, we show that every <span>(K_4)</span>-minor free graph <i>G</i> with maximum degree <span>(Delta (G)ge 3)</span> satisfies <span>(chi ^prime _{inj}(G)le 2Delta (G)+1)</span>.</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141255081","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}
Pub Date : 2024-06-01DOI: 10.1007/s00373-024-02802-8
Yue Ma, Xinmin Hou, Jun Gao
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.
狄拉克(Proc Lond Math Soc (3) 2:69-81,1952)证明了最小度大于 k 的每个阶为 (n>2k+1)的连通图都包含一条长度至少为 (2k+1)的路径。本文给出了狄拉克定理的超图扩展:给定正整数 n、k 和 r,设 H 是一个连通的 n 顶点 r 图,其中没有长度为 (2k+1)的 Berge 路径。(1) 如果(k> rge 4) 并且(n>2k+1),那么(delta _1(H)le left( {begin{array}{c}kr-1end{array}right) )。此外,存在超图 (S'_r(n,k), S_r(n,k)) 和 (S(sK_{k+1}^{(r)}、1)),使得当且仅当(S'_r(n,k)subseteq Hsubseteq S_r(n,k))或者(Hcong S(sK_{k+1}^{(r)},1)) 时,相等关系成立;(2) 如果 (kge rge 2) and (n>2k(r-1)), then (delta _1(H)le left( {begin{array}{c}k r-1end{array}}right)).作为(1)的应用,我们给出了一个比 Bermond 等人给出的 Berge Hamiltonian 循环的 Dirac 型结果(Hypergraphes Hamiltoniens.In:Problémes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976).Colloq.Internat.CNRS, vol. 260, pp.CNRS, Paris, 1976)或克莱门斯等人(Electron Notes Discrete Math 54:181-186, 2016)分别。
{"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":"<p>Dirac (Proc Lond Math Soc (3) 2:69–81, 1952) proved that every connected graph of order <span>(n>2k+1)</span> with minimum degree more than <i>k</i> contains a path of length at least <span>(2k+1)</span>. In this article, we give a hypergraph extension of Dirac’s theorem: Given positive integers <i>n</i>, <i>k</i> and <i>r</i>, let <i>H</i> be a connected <i>n</i>-vertex <i>r</i>-graph with no Berge path of length <span>(2k+1)</span>. (1) If <span>(k> rge 4)</span> and <span>(n>2k+1)</span>, then <span>(delta _1(H)le left( {begin{array}{c}k r-1end{array}}right) )</span>. Furthermore, there exist hypergraphs <span>(S'_r(n,k), S_r(n,k))</span> and <span>(S(sK_{k+1}^{(r)},1))</span> such that the equality holds if and only if <span>(S'_r(n,k)subseteq Hsubseteq S_r(n,k))</span> or <span>(Hcong S(sK_{k+1}^{(r)},1))</span>; (2) If <span>(kge rge 2)</span> and <span>(n>2k(r-1))</span>, then <span>(delta _1(H)le left( {begin{array}{c}k r-1end{array}}right) )</span>. 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.</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":null,"pages":null},"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}
Pub Date : 2024-05-30DOI: 10.1007/s00373-024-02803-7
Houmem Belkhechine, Cherifa Ben Salha, Rim Romdhane
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.
我们只考虑有限结构。对于每一个完全有序集合 V 和一个子集 P,我们通过反转 P, i 来关联从传递锦标赛({underline{V}}:=(V, {(x,y) in V times V: x < y }))通过反转 P 得到,即e.,通过反转弧(x, y),使得({x, y}in P )。如果 (|cup P| = 2|P|) 子集 P 是((cup P) 的)配对,如果 (|cup P| = 2|P|-1) 子集 P 是((cup P) 的)准配对;如果(cup P) 的无非数区间是图((cup P, P))的连接成分的联合,那么它就是不可还原的。在本文中,我们考虑配对和准配对与锦标赛的关系。我们建立了配对(或准配对)的不可还原性与其底层锦标赛在模块分解下的不可分解性之间的密切关系。例如,给定大小至少为 6 的完全有序集合 V 的配对 P,当且仅当锦标赛 (textrm{Inv}({underline{V}}, P))是不可分解的,配对 P 才是不可还原的。这是一个更一般的结果的结果,它描述了通过反转配对从反式锦标赛得到的不可分解锦标赛的特征。我们在准配对的情况下也得到了类似的结果。
{"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":"<p>We only consider finite structures. With every totally ordered set <i>V</i> and a subset <i>P</i> of <span>(left( {begin{array}{c}V 2end{array}}right) )</span>, we associate the underlying tournament <span>(textrm{Inv}({underline{V}}, P))</span> obtained from the transitive tournament <span>({underline{V}}:=(V, {(x,y) in V times V: x < y }))</span> by reversing <i>P</i>, i.e., by reversing the arcs (<i>x</i>, <i>y</i>) such that <span>({x,y} in P)</span>. The subset <i>P</i> is a pairing (of <span>(cup P)</span>) if <span>(|cup P| = 2|P|)</span>, a quasi-pairing (of <span>(cup P)</span>) if <span>(|cup P| = 2|P|-1)</span>; it is irreducible if no nontrivial interval of <span>(cup P)</span> is a union of connected components of the graph <span>((cup P, P))</span>. 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 <i>P</i> of a totally ordered set <i>V</i> of size at least 6, the pairing <i>P</i> is irreducible if and only if the tournament <span>(textrm{Inv}({underline{V}}, P))</span> 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.</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":null,"pages":null},"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}
Pub Date : 2024-05-30DOI: 10.1007/s00373-024-02798-1
Elliot Krop, Aryan Mittal, Michael C. Wigal
The cordiality game is played on a graph G by two players, Admirable (A) and Impish (I), who take turns selecting unlabeled vertices of G. Admirable labels the selected vertices by 0 and Impish by 1, and the resulting label on any edge is the sum modulo 2 of the labels of the vertices incident to that edge. The two players have opposite goals: Admirable attempts to minimize the number of edges with different labels as much as possible while Impish attempts to maximize this number. When both Admirable and Impish play their optimal games, we define the game cordiality number, (c_g(G)), as the absolute difference between the number of edges labeled zero and one. Let (P_n) be the path on n vertices. We show (c_g(P_n)le frac{n-3}{3}) when (n equiv 0 pmod 3), (c_g(P_n)le frac{n-1}{3}) when (n equiv 1 pmod 3), and (c_g(P_n)le frac{n+1}{3}) when (n equiv 2pmod 3). Furthermore, we show a similar bound, (c_g(T) le frac{|T|}{2}) holds for any tree T.
{"title":"The Cordiality Game and the Game Cordiality Number","authors":"Elliot Krop, Aryan Mittal, Michael C. Wigal","doi":"10.1007/s00373-024-02798-1","DOIUrl":"https://doi.org/10.1007/s00373-024-02798-1","url":null,"abstract":"<p>The <i>cordiality game</i> is played on a graph <i>G</i> by two players, Admirable (A) and Impish (I), who take turns selecting unlabeled vertices of <i>G</i>. Admirable labels the selected vertices by 0 and Impish by 1, and the resulting label on any edge is the sum modulo 2 of the labels of the vertices incident to that edge. The two players have opposite goals: Admirable attempts to minimize the number of edges with different labels as much as possible while Impish attempts to maximize this number. When both Admirable and Impish play their optimal games, we define the <i>game cordiality number</i>, <span>(c_g(G))</span>, as the absolute difference between the number of edges labeled zero and one. Let <span>(P_n)</span> be the path on <i>n</i> vertices. We show <span>(c_g(P_n)le frac{n-3}{3})</span> when <span>(n equiv 0 pmod 3)</span>, <span>(c_g(P_n)le frac{n-1}{3})</span> when <span>(n equiv 1 pmod 3)</span>, and <span>(c_g(P_n)le frac{n+1}{3})</span> when <span>(n equiv 2pmod 3)</span>. Furthermore, we show a similar bound, <span>(c_g(T) le frac{|T|}{2})</span> holds for any tree <i>T</i>.</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141194201","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}
Pub Date : 2024-05-30DOI: 10.1007/s00373-024-02800-w
Jie Chen, Cai-Xia Wang, Yi-Ping Liang, Shou-Jun Xu
In an isolate-free graph G, a subset S of vertices is a semitotal dominating set of G if it is a dominating set of G and every vertex in S is within distance 2 of another vertex of S. The semitotal domination number of G, denoted by (gamma _{t2}(G)), is the minimum cardinality of a semitotal dominating set in G. Zhu et al. (Gr Combin 33, 1119–1130, 2017) proved that if (Gnotin {K_4,N_2}) is a connected claw-free cubic graph of order n, then (gamma _{t2}(G)le frac{n}{3}), which is sharp. They proposed the problem of characterizing the extremal graphs. We completely solve this problem. There are ten classes of graphs, three of which are infinite families of graphs.
在无孤立图 G 中,如果顶点子集 S 是 G 的支配集,且 S 中的每个顶点与 S 中另一个顶点的距离都在 2 以内,则该顶点子集 S 是 G 的半总支配集。G 的半总支配数用 (gamma _{t2}(G)) 表示,它是 G 中半总支配集的最小卡片度。Zhu 等人(Gr Combin 33, 1119-1130, 2017)证明了如果 (Gnotin {K_4,N_2}) 是阶数为 n 的连通无爪立方图,那么 (gamma _{t2}(G)le frac{n}{3}) 是尖锐的。他们提出了极值图的特征问题。我们完全解决了这个问题。有十类图,其中三类是无限图族。
{"title":"A Characterization of Graphs with Semitotal Domination Number One-Third Their Order","authors":"Jie Chen, Cai-Xia Wang, Yi-Ping Liang, Shou-Jun Xu","doi":"10.1007/s00373-024-02800-w","DOIUrl":"https://doi.org/10.1007/s00373-024-02800-w","url":null,"abstract":"<p>In an isolate-free graph <i>G</i>, a subset <i>S</i> of vertices is a <i>semitotal dominating set</i> of <i>G</i> if it is a dominating set of <i>G</i> and every vertex in <i>S</i> is within distance 2 of another vertex of <i>S</i>. The <i>semitotal domination number</i> of <i>G</i>, denoted by <span>(gamma _{t2}(G))</span>, is the minimum cardinality of a semitotal dominating set in <i>G</i>. Zhu et al. (Gr Combin 33, 1119–1130, 2017) proved that if <span>(Gnotin {K_4,N_2})</span> is a connected claw-free cubic graph of order <i>n</i>, then <span>(gamma _{t2}(G)le frac{n}{3})</span>, which is sharp. They proposed the problem of characterizing the extremal graphs. We completely solve this problem. There are ten classes of graphs, three of which are infinite families of graphs.</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141194205","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}
Pub Date : 2024-05-23DOI: 10.1007/s00373-024-02777-6
Vadim E. Levit, David Tankus
{"title":"Recognizing $$mathbf {W_2}$$ Graphs","authors":"Vadim E. Levit, David Tankus","doi":"10.1007/s00373-024-02777-6","DOIUrl":"https://doi.org/10.1007/s00373-024-02777-6","url":null,"abstract":"","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141107697","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}
Pub Date : 2024-05-21DOI: 10.1007/s00373-024-02792-7
Dermot McCarthy, Mason Springfield
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).
{"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":"<p>Let <span>(k ge 2)</span> be an even integer. Let <i>q</i> be a prime power such that <span>(q equiv k+1 (text {mod},,2k))</span>. We define the <i>k-th power Paley digraph</i> of order <i>q</i>, <span>(G_k(q))</span>, as the graph with vertex set <span>(mathbb {F}_q)</span> where <span>(a rightarrow b)</span> is an edge if and only if <span>(b-a)</span> is a <i>k</i>-th power residue. This generalizes the (<span>(k=2)</span>) Paley Tournament. We provide a formula, in terms of finite field hypergeometric functions, for the number of transitive subtournaments of order four contained in <span>(G_k(q))</span>, <span>(mathcal {K}_4(G_k(q)))</span>, which holds for all <i>k</i>. We also provide a formula, in terms of Jacobi sums, for the number of transitive subtournaments of order three contained in <span>(G_k(q))</span>, <span>(mathcal {K}_3(G_k(q)))</span>. In both cases, we give explicit determinations of these formulae for small <i>k</i>. We show that zero values of <span>(mathcal {K}_4(G_k(q)))</span> (resp. <span>(mathcal {K}_3(G_k(q)))</span>) yield lower bounds for the multicolor directed Ramsey numbers <span>(R_{frac{k}{2}}(4)=R(4,4,ldots ,4))</span> (resp. <span>(R_{frac{k}{2}}(3))</span>). We state explicitly these lower bounds for <span>(kle 10)</span> and compare to known bounds, showing improvement for <span>(R_2(4))</span> and <span>(R_3(3))</span>. Combining with known multiplicative relations we give improved lower bounds for <span>(R_{t}(4))</span>, for all <span>(tge 2)</span>, and for <span>(R_{t}(3))</span>, for all <span>(t ge 3)</span>.</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":null,"pages":null},"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}