An edge coloring of hypergraph H is a function such that holds for any pair of intersecting edges . The minimum number of colors in edge colorings of H is called the chromatic index of H and is ...An edge coloring of hypergraph H is a function such that holds for any pair of intersecting edges . The minimum number of colors in edge colorings of H is called the chromatic index of H and is denoted by . Erdös, Faber and Lovász proposed a famous conjecture that holds for any loopless linear hypergraph H with n vertices. In this paper, we show that is true for gap-restricted hypergraphs. Our result extends a result of Alesandroni in 2021.展开更多
The celebrated Erdos-Ko-Rado theorem states that given n≥2k,every intersecting k-uni-n-1 form hypergraph G on n vertices has at most(n-1 k-1)edges.This paper states spectral versions of the Erd6s-_Ko--Rado theorem:le...The celebrated Erdos-Ko-Rado theorem states that given n≥2k,every intersecting k-uni-n-1 form hypergraph G on n vertices has at most(n-1 k-1)edges.This paper states spectral versions of the Erd6s-_Ko--Rado theorem:let G be an intersecting k-uniform hypergraph on n vertices with n≥2k.Then,the sharp upper bounds for the spectral radius of Aα(G)and 2*(G)are presented,where Aα(G)=αD(G)+(1-α).A(G)is a convex linear combination of the degree diagonal tensor D(G)and the adjacency tensor A(G)for 0≤α<1,and Q^(*)(G)is the incidence Q-tensor,respectively.Furthermore,when n>2k,the extremal hypergraphs which attain the sharp upper bounds are characterized.The proof mainly relies on the Perron-Frobenius theorem for nonnegative tensor and the property of the maximizing connected hypergraphs.展开更多
Let p,q be two positive integers.The 3-graph F(p,q)is obtained from the complete 3-graph K_(p)^(3)by adding q new vertices and P_(q/2)new edges of the form vxy for which v∈V(K_p~3)and{x,y}are new vertices.It frequent...Let p,q be two positive integers.The 3-graph F(p,q)is obtained from the complete 3-graph K_(p)^(3)by adding q new vertices and P_(q/2)new edges of the form vxy for which v∈V(K_p~3)and{x,y}are new vertices.It frequently appears in many literatures on the Turán number or Turán density of hypergraphs.In this paper,we first construct a new class of r-graphs which can be regarded as a generalization of the 3-graph F(p,q),and prove that these r-graphs have the same Turán density under some situations.Moreover,we investigate the Turán density of the F(p,q)for small p,q and obtain some new bounds on their Turán densities.展开更多
A hypergraph H is an(n,m)-hypergraph if it contains n vertices and m hyperedges,where n≥1 and m≥0 are two integers.Let k be a positive integer and let L be a set of nonnegative integers.A hyper graph H is k-uniform ...A hypergraph H is an(n,m)-hypergraph if it contains n vertices and m hyperedges,where n≥1 and m≥0 are two integers.Let k be a positive integer and let L be a set of nonnegative integers.A hyper graph H is k-uniform if all its hyperedges have the same size k,and H is L-intersecting if the number of common vertices of every two hyperedges belongs to L.In this paper,we propose and investigate the problem of estimating the maximum k among all k-uniform L-intersecting(n,m)-hypergraphs for fixed n,m and L.We will provide some tight upper and lower bounds on k in terms of n,m and L.展开更多
Let H=(V,E)be a hypergraph,where V is a set of vertices and E is a set of non-empty subsets of V called edges.If all edges of H have the same cardinality r,then H is an r-uniform hypergraph;if E consists of all r-subs...Let H=(V,E)be a hypergraph,where V is a set of vertices and E is a set of non-empty subsets of V called edges.If all edges of H have the same cardinality r,then H is an r-uniform hypergraph;if E consists of all r-subsets of V,then H is a complete r-uniform hypergraph,denoted by K_(n)^(r),where n=|V|.A hypergraph H′=(V′,E′)is called a subhypergraph of H=(V,E)if V′⊆V and E′⊆E.The edge-connectivity of a hypergraph H is the cardinality of a minimum edge set F⊆E such that H−F is not connected,where H−F=(V,E\F).An r-uniform hypergraph H=(V,E)is k-edge-maximal if every subhypergraph of H has edge-connectivity at most k,but for any edge e∈E(K_(n)^(r))\E(H),H+e contains at least one subhypergraph with edge-connectivity at least k+1.Let k and r be integers with k≥2 and r≥2,and let t=t(k,r)be the largest integer such that(t−1 r−1)≤k.That is,t is the integer satisfying(t−1 r−1)≤k<(t r−1).We prove that if H is an r-uniform k-edge-maximal hypergraph such that n=|V(H)|≥t,then(i)|E(H)|≤(t r)+(n−t)k,and this bound is best possible;(ii)|E(H)|≥(n−1)k−((t−1)k−(t r))[n/t],and this bound is best possible.展开更多
Acyclic hypergraphs are analogues of forests in graphs. They arevery useful in the design of databases. The number of distinct acyclic uniform hypergraphs with n labeled vertices is studied. With the aid of the princi...Acyclic hypergraphs are analogues of forests in graphs. They arevery useful in the design of databases. The number of distinct acyclic uniform hypergraphs with n labeled vertices is studied. With the aid of the principle of inclusion-exclusion, two formulas are presented. One is the explicit formula for strict (d)-connected acyclic hypergraphs, the other is the recurrence formula for linear acyclic hypergraphs.展开更多
In this paper,we investigate a generalization of graph decomposition,called hypergraph decomposition.We show that a decomposition of a 3-uniform hypergraph K<sub>v</sub><sup>3</sup>into a speci...In this paper,we investigate a generalization of graph decomposition,called hypergraph decomposition.We show that a decomposition of a 3-uniform hypergraph K<sub>v</sub><sup>3</sup>into a special kind of hypergraph K<sub>4</sub><sup>3</sup>-e exists if and only if v≡0,1,2(mod 9)and v≥9.展开更多
Hypergraphs are the most general structures in discrete mathematics. Acyclic hypergraphs have been proved very useful in relational databases. New systems of axioms for paths, connectivity and cycles of hypergraphs ar...Hypergraphs are the most general structures in discrete mathematics. Acyclic hypergraphs have been proved very useful in relational databases. New systems of axioms for paths, connectivity and cycles of hypergraphs are constructed. The systems suit the structure properties of relational databases. The concepts of pseudo cycles and essential cycles of hypergraphs are introduced. They are relative to each other. Whether a family of cycles of a hypergraph is dependent or independent is defined. An enumeration formula for the maximum number of independent essential cycles of a hypergraph is given.展开更多
Let Η be a hypergraph with n vertices. Suppose that di,¢/2,...,dn are degrees of the vert ices of Η. The t-th graph entropy based on degrees of H is defined as Id^t(Η)=-n∑i=1(di^t/∑j=1^ndj^t^nlogdi^t/∑j=1^ndj^t...Let Η be a hypergraph with n vertices. Suppose that di,¢/2,...,dn are degrees of the vert ices of Η. The t-th graph entropy based on degrees of H is defined as Id^t(Η)=-n∑i=1(di^t/∑j=1^ndj^t^nlogdi^t/∑j=1^ndj^t^n)=log(n∑i=1di^t)-n∑i=1(di^t/∑j=1^ndj^tlogdi^t), where t is a real number and the logarithm is taken to the base two. In this paper we obtain upper and lower bounds of Id^t(Η) for t = 1, when Η is among all uniform super trees, unicyclic uniform hypergraphs and bicyclic uniform hypergraphs, respectively.展开更多
In this paper,we study the adjacency and signless Laplacian tensors of cored hypergraphs and power hypergraphs.We investigate the properties of their adjacency and signless Laplacian H-eigenvalues.Especially,we find o...In this paper,we study the adjacency and signless Laplacian tensors of cored hypergraphs and power hypergraphs.We investigate the properties of their adjacency and signless Laplacian H-eigenvalues.Especially,we find out the largest H-eigenvalues of adjacency and signless Laplacian tensors for uniform squids.We also compute the H-spectra of sunflowers and some numerical results are reported for the H-spectra.展开更多
Cyclic hypergraphs are the analogue of cyclic graphs. The unicycle hypergraph whose cyclomatic number equals to one is the most elemental cyclic hypergraph. In this paper the maximum size of unicycle hypergraphs is st...Cyclic hypergraphs are the analogue of cyclic graphs. The unicycle hypergraph whose cyclomatic number equals to one is the most elemental cyclic hypergraph. In this paper the maximum size of unicycle hypergraphs is studied. It is proved a piecewise linear function of the order and edge size of the hypergraph.展开更多
The explicit formula for (k+l)-uniform linear acyclic hypergraphs and the counting series for unlabeled (k +1)-uniform linear acyclic hypergraphs are obtained.
Let G be a connected hypergraph with even uniformity,which contains cut vertices.Then G is the coalescence of two nontrivial connected sub-hypergraphs(called branches)at a cut vertex.Let A(G)be the adjacency tensor of...Let G be a connected hypergraph with even uniformity,which contains cut vertices.Then G is the coalescence of two nontrivial connected sub-hypergraphs(called branches)at a cut vertex.Let A(G)be the adjacency tensor of G.The least H-eigenvalue of A(G)refers to the least real eigenvalue of A(G)associated with a real eigenvector.In this paper,we obtain a perturbation result on the least H-eigenvalue of A(G)when a branch of G attached at one vertex is relocated to another vertex,and characterize the unique hypergraph whose least H-eigenvalue attains the minimum among all hypergraphs in a certain class of hypergraphs which contain a fixed connected hypergraph.展开更多
For 0≤α<1,theα-spectral radius of an r-uniform hypergraph G is the spectral radius of A_(α)(G)=αD(G)+(1-α)A(G),where D(G)and A(G)are the diagonal tensor of degrees and adjacency tensor of G,respectively.In th...For 0≤α<1,theα-spectral radius of an r-uniform hypergraph G is the spectral radius of A_(α)(G)=αD(G)+(1-α)A(G),where D(G)and A(G)are the diagonal tensor of degrees and adjacency tensor of G,respectively.In this paper,we show the perturbation ofα-spectral radius by contracting an edge.Then we determine the unique unicyclic hypergraph with the maximumα-spectral radius among all r-uniform unicyclic hypergraphs with fixed diameter.We also determine the unique unicyclic hypergraph with the maximumα-spectral radius among all r-uniform unicyclic hypergraphs with given number of pendant edges.展开更多
Let ■ be a k-uniform hypergraph on n vertices with degree sequence △= d1≥…≥ dn =δ. In this paper, in terms of degree di , we give some upper bounds for the Z-spectral radius of the signless Laplacian tensor (Q(...Let ■ be a k-uniform hypergraph on n vertices with degree sequence △= d1≥…≥ dn =δ. In this paper, in terms of degree di , we give some upper bounds for the Z-spectral radius of the signless Laplacian tensor (Q(■)) of ■. Some examples are given to show the efficiency of these bounds.展开更多
Let H=(V,E)be an n-balanced k-partite k-graph with partition classes V1,...,Vk.Suppose for every legal(k-1)-tuple f contained in V\V1 and for every legal(k-1)-tuple g contained in V\Vk such that f∪g■E(H),we have d(f...Let H=(V,E)be an n-balanced k-partite k-graph with partition classes V1,...,Vk.Suppose for every legal(k-1)-tuple f contained in V\V1 and for every legal(k-1)-tuple g contained in V\Vk such that f∪g■E(H),we have d(f)+d(g)≥n+1.In this paper,we prove that under this condition H must have a perfect matching.Another result of this paper is about the perfect matching in 3-uniform hm-bipartite hypergraphs.Let G be a 3-uniform hm-bipartite hypergraph with one of whose sides V1 has the size n,the another side V2 has size 2 n.If for all the legal 2-tuple f with|f∩V1|=1 and for all the legal 2-tuple g with|g∩V1|=0,we have d(f)≥n-2 and d(g)>n/2,then G has a perfect matching.展开更多
The edge-connectivity of a graph or a hypergraph is defined as the minimum number of edges whose removal renders the graph or hypergraph disconnected.A graph or hypergraph is called maximally edge-connected if the edg...The edge-connectivity of a graph or a hypergraph is defined as the minimum number of edges whose removal renders the graph or hypergraph disconnected.A graph or hypergraph is called maximally edge-connected if the edge-connectivity equals its minimum degree.In this paper,we show that some classical sufficient conditions for graphs to be maximally edge-connected can be generalized to hypergraphs.展开更多
Obtaining Turán densities of hypergraph through determining Lagrangian densities have been studied actively.Sidorenko showed that the Lagrangian density of an r-uniform hypergraph equals the Turán density of...Obtaining Turán densities of hypergraph through determining Lagrangian densities have been studied actively.Sidorenko showed that the Lagrangian density of an r-uniform hypergraph equals the Turán density of the extension of this hypergraph.Indeed,Lagrangian density and Turán density are the same for a large class of hypergraphs such as hypergraphs covering pairs.Let Pr,sbe the r-uniform hypergraph with two edges intersecting at s vertices.Sidorenko determined the Lagrangian densities of Pr,sfor r=3 and s=1 or 2,or r=4 and s=2 or 3.Hefetz and Keevash determined for r=3 and s=0,and Bene Watts,Norin and Yepremyan determined for r≥4 and s=0.The cases r=5 and s=4 or r=6 and s=5 were determined by Norin and Yepremyan.Jenssen showed for r=5,6 or 7,and s=3,4 or 5 respectively.The cases r=4 and s=1 or r=5 and s=2 were determined by Jiang,Peng and Wu,and Hu,Peng and Wu respectively.For r=5,the only remaining case is s=1.In this paper,we determine for this case.展开更多
In [1] the concepts of paths and cycles of a hypergraph were introduced. In this paper, we give the concepts for bipartite hypergraph and Hamiltonian paths and cycles of a hypergraph, and prove that the complete bipar...In [1] the concepts of paths and cycles of a hypergraph were introduced. In this paper, we give the concepts for bipartite hypergraph and Hamiltonian paths and cycles of a hypergraph, and prove that the complete bipartite 3-hypergraph with q vertices in earh part is Hamiltonian decomposable where q is a prime.展开更多
文摘An edge coloring of hypergraph H is a function such that holds for any pair of intersecting edges . The minimum number of colors in edge colorings of H is called the chromatic index of H and is denoted by . Erdös, Faber and Lovász proposed a famous conjecture that holds for any loopless linear hypergraph H with n vertices. In this paper, we show that is true for gap-restricted hypergraphs. Our result extends a result of Alesandroni in 2021.
基金the National Natural Science Foundation of China(Nos.11971311,11531001)the Montenegrin-Chinese Science and Technology Cooperation Project(No.3-12).
文摘The celebrated Erdos-Ko-Rado theorem states that given n≥2k,every intersecting k-uni-n-1 form hypergraph G on n vertices has at most(n-1 k-1)edges.This paper states spectral versions of the Erd6s-_Ko--Rado theorem:let G be an intersecting k-uniform hypergraph on n vertices with n≥2k.Then,the sharp upper bounds for the spectral radius of Aα(G)and 2*(G)are presented,where Aα(G)=αD(G)+(1-α).A(G)is a convex linear combination of the degree diagonal tensor D(G)and the adjacency tensor A(G)for 0≤α<1,and Q^(*)(G)is the incidence Q-tensor,respectively.Furthermore,when n>2k,the extremal hypergraphs which attain the sharp upper bounds are characterized.The proof mainly relies on the Perron-Frobenius theorem for nonnegative tensor and the property of the maximizing connected hypergraphs.
基金the National Natural Science Foundation of China(No.12171089)。
文摘Let p,q be two positive integers.The 3-graph F(p,q)is obtained from the complete 3-graph K_(p)^(3)by adding q new vertices and P_(q/2)new edges of the form vxy for which v∈V(K_p~3)and{x,y}are new vertices.It frequently appears in many literatures on the Turán number or Turán density of hypergraphs.In this paper,we first construct a new class of r-graphs which can be regarded as a generalization of the 3-graph F(p,q),and prove that these r-graphs have the same Turán density under some situations.Moreover,we investigate the Turán density of the F(p,q)for small p,q and obtain some new bounds on their Turán densities.
基金Supported by National Natural Science Foundation of China(Grant Nos.12242111,12131013,12171393,12071370,71973103,U1803263,11601430)Natural Science Foundation of Shaanxi Province(Grant Nos.2021JM-040,2020JQ-099)+2 种基金Shaanxi Fundamental Science Research Project for Mathematics and Physics(Grant No.22JSZ009)Guangdong Basic and Applied Basic Research Foundation(Grant Nos.2023A1515030208,2022A1515010899)the Fundamental Research Funds for the Central Universities。
文摘A hypergraph H is an(n,m)-hypergraph if it contains n vertices and m hyperedges,where n≥1 and m≥0 are two integers.Let k be a positive integer and let L be a set of nonnegative integers.A hyper graph H is k-uniform if all its hyperedges have the same size k,and H is L-intersecting if the number of common vertices of every two hyperedges belongs to L.In this paper,we propose and investigate the problem of estimating the maximum k among all k-uniform L-intersecting(n,m)-hypergraphs for fixed n,m and L.We will provide some tight upper and lower bounds on k in terms of n,m and L.
基金supported by the National Natural Science Foundation of China(Nos.11861066,11531011)Tianshan Youth Project of Xinjiang(2018Q066)。
文摘Let H=(V,E)be a hypergraph,where V is a set of vertices and E is a set of non-empty subsets of V called edges.If all edges of H have the same cardinality r,then H is an r-uniform hypergraph;if E consists of all r-subsets of V,then H is a complete r-uniform hypergraph,denoted by K_(n)^(r),where n=|V|.A hypergraph H′=(V′,E′)is called a subhypergraph of H=(V,E)if V′⊆V and E′⊆E.The edge-connectivity of a hypergraph H is the cardinality of a minimum edge set F⊆E such that H−F is not connected,where H−F=(V,E\F).An r-uniform hypergraph H=(V,E)is k-edge-maximal if every subhypergraph of H has edge-connectivity at most k,but for any edge e∈E(K_(n)^(r))\E(H),H+e contains at least one subhypergraph with edge-connectivity at least k+1.Let k and r be integers with k≥2 and r≥2,and let t=t(k,r)be the largest integer such that(t−1 r−1)≤k.That is,t is the integer satisfying(t−1 r−1)≤k<(t r−1).We prove that if H is an r-uniform k-edge-maximal hypergraph such that n=|V(H)|≥t,then(i)|E(H)|≤(t r)+(n−t)k,and this bound is best possible;(ii)|E(H)|≥(n−1)k−((t−1)k−(t r))[n/t],and this bound is best possible.
基金This work was supported by the National Natural Science Foundation of China (Grant No. 19831080) .
文摘Acyclic hypergraphs are analogues of forests in graphs. They arevery useful in the design of databases. The number of distinct acyclic uniform hypergraphs with n labeled vertices is studied. With the aid of the principle of inclusion-exclusion, two formulas are presented. One is the explicit formula for strict (d)-connected acyclic hypergraphs, the other is the recurrence formula for linear acyclic hypergraphs.
基金This work was supported by the National Natural Science Foundation of China(Grant No.10371002)
文摘In this paper,we investigate a generalization of graph decomposition,called hypergraph decomposition.We show that a decomposition of a 3-uniform hypergraph K<sub>v</sub><sup>3</sup>into a special kind of hypergraph K<sub>4</sub><sup>3</sup>-e exists if and only if v≡0,1,2(mod 9)and v≥9.
文摘Hypergraphs are the most general structures in discrete mathematics. Acyclic hypergraphs have been proved very useful in relational databases. New systems of axioms for paths, connectivity and cycles of hypergraphs are constructed. The systems suit the structure properties of relational databases. The concepts of pseudo cycles and essential cycles of hypergraphs are introduced. They are relative to each other. Whether a family of cycles of a hypergraph is dependent or independent is defined. An enumeration formula for the maximum number of independent essential cycles of a hypergraph is given.
基金Supported by NSFC(Grant Nos.11531011,11671320,11601431,11871034 and U1803263)the China Postdoctoral Science Foundation(Grant No.2016M600813)the Natural Science Foundation of Shaanxi Province(Grant No.2017JQ1019)
文摘Let Η be a hypergraph with n vertices. Suppose that di,¢/2,...,dn are degrees of the vert ices of Η. The t-th graph entropy based on degrees of H is defined as Id^t(Η)=-n∑i=1(di^t/∑j=1^ndj^t^nlogdi^t/∑j=1^ndj^t^n)=log(n∑i=1di^t)-n∑i=1(di^t/∑j=1^ndj^tlogdi^t), where t is a real number and the logarithm is taken to the base two. In this paper we obtain upper and lower bounds of Id^t(Η) for t = 1, when Η is among all uniform super trees, unicyclic uniform hypergraphs and bicyclic uniform hypergraphs, respectively.
基金the National Natural Science Foundation of China(No.11271221)the Specialized Research Fund for State Key Laboratories.
文摘In this paper,we study the adjacency and signless Laplacian tensors of cored hypergraphs and power hypergraphs.We investigate the properties of their adjacency and signless Laplacian H-eigenvalues.Especially,we find out the largest H-eigenvalues of adjacency and signless Laplacian tensors for uniform squids.We also compute the H-spectra of sunflowers and some numerical results are reported for the H-spectra.
文摘Cyclic hypergraphs are the analogue of cyclic graphs. The unicycle hypergraph whose cyclomatic number equals to one is the most elemental cyclic hypergraph. In this paper the maximum size of unicycle hypergraphs is studied. It is proved a piecewise linear function of the order and edge size of the hypergraph.
基金This work was supported by the National Natural Science Foundation of China (Grant No. 19771040)and the Natural Science Foundation of Guangdong Province.
文摘The explicit formula for (k+l)-uniform linear acyclic hypergraphs and the counting series for unlabeled (k +1)-uniform linear acyclic hypergraphs are obtained.
基金This work was supported by the National Natural Science Foundation of China(Grant Nos.11871073,11771016).
文摘Let G be a connected hypergraph with even uniformity,which contains cut vertices.Then G is the coalescence of two nontrivial connected sub-hypergraphs(called branches)at a cut vertex.Let A(G)be the adjacency tensor of G.The least H-eigenvalue of A(G)refers to the least real eigenvalue of A(G)associated with a real eigenvector.In this paper,we obtain a perturbation result on the least H-eigenvalue of A(G)when a branch of G attached at one vertex is relocated to another vertex,and characterize the unique hypergraph whose least H-eigenvalue attains the minimum among all hypergraphs in a certain class of hypergraphs which contain a fixed connected hypergraph.
基金Supported by the National Nature Science Foundation of China(Grant Nos.11871329,11971298)。
文摘For 0≤α<1,theα-spectral radius of an r-uniform hypergraph G is the spectral radius of A_(α)(G)=αD(G)+(1-α)A(G),where D(G)and A(G)are the diagonal tensor of degrees and adjacency tensor of G,respectively.In this paper,we show the perturbation ofα-spectral radius by contracting an edge.Then we determine the unique unicyclic hypergraph with the maximumα-spectral radius among all r-uniform unicyclic hypergraphs with fixed diameter.We also determine the unique unicyclic hypergraph with the maximumα-spectral radius among all r-uniform unicyclic hypergraphs with given number of pendant edges.
基金Science and Technology Foundation of Guizhou Province (Qian Ke He Ji Chu [2016]1161)Natural Science Foundation of Guizhou Province (Qian Jiao He KY [2016]255)+9 种基金Doctoral Scientific Research Foundation of Zunyi Normal College (BS[2015]09)Yanmin Liu was supported by the National Natural Science Foundations of China (Grant No. 71461027)Science and Technology Talent Training Object of Guizhou Province Outstanding Youth (Qian Ke He Ren Zi [2015]06)Natural Science Foundation of Guizhou Province (Qian Jiao He KY [2014]295), 2013. 2014, 2015 Zunyi 15851 Talents Elite Project FundingZunyi Innovative Talent Team (Zunyi KH (2015)38)Junkang Tian was supported by the Natural Science Foundation of Guizhou Province (Qian Jiao He KY [2015]451)Science and Technology Foundation of Guizhou Province (Qian Ke He J Zi [2015]2147)Xianghu Liu was supported by the Guizhou Province Department of Education Fund (KY[2015]391,[2016]046)Guizhou Province Department of Education Teaching Reform Project ([2015]337)Guizhou Province Science and Technology Fund (qian Ke He Ji Chu [2016]1160).
文摘Let ■ be a k-uniform hypergraph on n vertices with degree sequence △= d1≥…≥ dn =δ. In this paper, in terms of degree di , we give some upper bounds for the Z-spectral radius of the signless Laplacian tensor (Q(■)) of ■. Some examples are given to show the efficiency of these bounds.
基金supported in part by the National Natural Science Foundation of China (No. 61373019)
文摘Let H=(V,E)be an n-balanced k-partite k-graph with partition classes V1,...,Vk.Suppose for every legal(k-1)-tuple f contained in V\V1 and for every legal(k-1)-tuple g contained in V\Vk such that f∪g■E(H),we have d(f)+d(g)≥n+1.In this paper,we prove that under this condition H must have a perfect matching.Another result of this paper is about the perfect matching in 3-uniform hm-bipartite hypergraphs.Let G be a 3-uniform hm-bipartite hypergraph with one of whose sides V1 has the size n,the another side V2 has size 2 n.If for all the legal 2-tuple f with|f∩V1|=1 and for all the legal 2-tuple g with|g∩V1|=0,we have d(f)≥n-2 and d(g)>n/2,then G has a perfect matching.
基金This research was partially supported by the National Natural Science Foundation of China(Nos.11571222 and 11871329).
文摘The edge-connectivity of a graph or a hypergraph is defined as the minimum number of edges whose removal renders the graph or hypergraph disconnected.A graph or hypergraph is called maximally edge-connected if the edge-connectivity equals its minimum degree.In this paper,we show that some classical sufficient conditions for graphs to be maximally edge-connected can be generalized to hypergraphs.
基金Supported by National Natural Science Foundation of China(Grant Nos.11901193 and 11931002)National Natural Science Foundation of Hunan Province,China(Grant No.2019JJ50364)the Construct Program of the Key Discipline in Hu’nan Province。
文摘Obtaining Turán densities of hypergraph through determining Lagrangian densities have been studied actively.Sidorenko showed that the Lagrangian density of an r-uniform hypergraph equals the Turán density of the extension of this hypergraph.Indeed,Lagrangian density and Turán density are the same for a large class of hypergraphs such as hypergraphs covering pairs.Let Pr,sbe the r-uniform hypergraph with two edges intersecting at s vertices.Sidorenko determined the Lagrangian densities of Pr,sfor r=3 and s=1 or 2,or r=4 and s=2 or 3.Hefetz and Keevash determined for r=3 and s=0,and Bene Watts,Norin and Yepremyan determined for r≥4 and s=0.The cases r=5 and s=4 or r=6 and s=5 were determined by Norin and Yepremyan.Jenssen showed for r=5,6 or 7,and s=3,4 or 5 respectively.The cases r=4 and s=1 or r=5 and s=2 were determined by Jiang,Peng and Wu,and Hu,Peng and Wu respectively.For r=5,the only remaining case is s=1.In this paper,we determine for this case.
基金the National Natural Science Foundation of China (No.19831080).
文摘In [1] the concepts of paths and cycles of a hypergraph were introduced. In this paper, we give the concepts for bipartite hypergraph and Hamiltonian paths and cycles of a hypergraph, and prove that the complete bipartite 3-hypergraph with q vertices in earh part is Hamiltonian decomposable where q is a prime.