In this paper the authors generalize the classic random bipartite graph model, and define a model of the random bipartite multigraphs as follows：let m = m（n） be a positive integer-valued function on n and ζ（n,m;{...In this paper the authors generalize the classic random bipartite graph model, and define a model of the random bipartite multigraphs as follows：let m = m（n） be a positive integer-valued function on n and ζ（n,m;{pk}） the probability space consisting of all the labeled bipartite multigraphs with two vertex sets A ={a_1,a_2,...,a_n} and B = {b_1,b_2,...,b_m}, in which the numbers t_（ai）,b_j of the edges between any two vertices a_i∈A and b_j∈ B are identically distributed independent random variables with distribution P{t_（ai）,b_j=k}=pk,k=0,1,2,...,where pk ≥0 and ∞Σk=0 pk=1. They obtain that X_（c,d,A）, the number of vertices in A with degree between c and d of G_（n,m）∈ζ（n, m;{pk}） has asymptotically Poisson distribution, and answer the following two questions about the space ζ（n,m;{pk}） with {pk} having geometric distribution, binomial distribution and Poisson distribution, respectively. Under which condition for {pk} can there be a function D（n） such that almost every random multigraph G_（n,m）∈ζ（n,m;{pk}） has maximum degree D（n）in A？ under which condition for {pk} has almost every multigraph G（n,m）∈ζ（n,m;{pk}） a unique vertex of maximum degree in A？展开更多
Let a, b, r be nonnegative integers with 1 ≤ a ≤ b and r ≥ 2. Let G be a graph of order n with n 〉（a＋2 b）（r（a＋b）-2）/b.In this paper, we prove that G is fractional ID-[a, b]-factor-critical if δ（G）≥bn/a...Let a, b, r be nonnegative integers with 1 ≤ a ≤ b and r ≥ 2. Let G be a graph of order n with n 〉（a＋2 b）（r（a＋b）-2）/b.In this paper, we prove that G is fractional ID-[a, b]-factor-critical if δ（G）≥bn/a＋2 b＋a（r-1）and ｜NG（x1） ∪ NG（x2） ∪…∪ NG（xr）｜ ≥（a＋b）n/（a＋2 b） for any independent subset {x1,x2,…,xr} in G. It is a generalization of Zhou et al.＇s previous result [Discussiones Mathematicae Graph Theory, 36： 409-418（2016）]in which r = 2 is discussed. Furthermore, we show that this result is best possible in some sense.展开更多
The harmonic index of a graph?G? is defined as where d(u) denotes the degree of a vertex u in G . In this work, we give another expression for the Harmonic index. Using this expression, we give the minimum value of th...The harmonic index of a graph?G? is defined as where d(u) denotes the degree of a vertex u in G . In this work, we give another expression for the Harmonic index. Using this expression, we give the minimum value of the harmonic index for any triangle-free graphs with order n and minimum degree δ ≥ k for k≤ n/2? and show the corresponding extremal graph is the complete graph.展开更多
In this paper we consider the random r-uniform r-partite hypergraph model H（n1, n2,…, nr; n, p） which consists of all the r-uniform r-partite hypergraphs with vertex partition {V1,V2,…, Vr} where ｜Vi｜ = ni = ni...In this paper we consider the random r-uniform r-partite hypergraph model H（n1, n2,…, nr; n, p） which consists of all the r-uniform r-partite hypergraphs with vertex partition {V1,V2,…, Vr} where ｜Vi｜ = ni = ni（n） （1 ≤ i ≤ r） are positive integer-valued functions on n with n1 ＋n2 ＋… ＋nr =n, and each r-subset containing exactly one element in Vi （1 ≤ i ≤ r） is chosen to be a hyperedge of Hp ∈H（n1,n2,…,nr;n,p） with probability p = p（n）, all choices being independent. Let △V1 = △V1 （H） and δv1 = δv1（H） be the maximum and minimum degree of vertices in V1 of H, respectively; Xd,V1 = Xd,V1 （H）, Yd,V1 = Yd,V1 （H）, Zd,V1 = Zd,V1 （H） and Zc,d,V1=Zc,d,V1 （H） be the number of vertices in V1 of H with degree d, at least d, at most d, and between c and d, respectively. In this paper we obtain that in the space H（n1, n2,…, nr; n,p）, Xd,V1, Yd,V1, Zd,V1, and Zc,d,V1all have asymptotically Poisson distributions. We also answer the following two questions. What is the range of p that there exists a function D（n） such that in the space H（n1, n2,…,nr; n, p）, lim n→＋∞ P（△v1 = D（n）） = 1？ What is the range of p such that a.e., Hp ∈ H（n1,n2,...,nr;n,p） has a unique vertex in V1 with degree Av1（Hp）？ Both answers are p = o（logn1/N）, where N = r ∏ i=2 ni. The corresponding problems on δv1（Hp） also are considered, and we obtained the answers are p ≤ （1＋o（1））（logn1/N） andp=o （log n1/N）, respectively.展开更多
In this work, we obtain good upper bounds for the diameter of any graph in terms of its minimum degree and its order, improving a classical theorem due to Erdos, Pach, Pollack and Tuza. We use these bounds in order to...In this work, we obtain good upper bounds for the diameter of any graph in terms of its minimum degree and its order, improving a classical theorem due to Erdos, Pach, Pollack and Tuza. We use these bounds in order to study hyperbolic graphs （in the Gromov sense）. To compute the hyperbolicity constant is an almost intractable problem, thus it is natural to try to bound it in terms of some parameters of the graph. Let H（n, δ0） be the set of graphs G with n vertices and minimum degree 50, and J（n, Δ） be the set of graphs G with n vertices and maximum degree A. We study the four following extremal problems on graphs： a（n,δ0） = min{δ（G） ｜ G ∈H（n, δ0）}, b（n, δ0） =- max{δ（G）｜ e ∈H（n, δ0）}, α（n, Δ） = min{δ（G） [ G ∈ J（n, Δ）} and β（n,Δ） = max{δ（G） ] G∈Π（n,Δ）}. In particular, we obtain bounds for b（n, δ0） and we compute the precise value of a（n, δ0）, α（n, Δ） and w（n, Δ） for all values of n, r0 and A, respectively.展开更多
The original version of the article was published in [1]. Unfortunately, the original version of this article contains a mistake： in Theorem 6.2 appears that β（n, △） = （n-△ ＋ 5）/4 but the correct statement is...The original version of the article was published in [1]. Unfortunately, the original version of this article contains a mistake： in Theorem 6.2 appears that β（n, △） = （n-△ ＋ 5）/4 but the correct statement is β（n, △） = （n -△ ＋ 4）/4. In this erratum we correct the theorem and give the correct proof.展开更多
A graph is said to be claw-free if it does not contain an induced subgraph isomorphic to K1,3. Let K4 be the graph obtained by removing exactly one edge from K4 and let k be an integer with k ≥ 2. We prove that if G ...A graph is said to be claw-free if it does not contain an induced subgraph isomorphic to K1,3. Let K4 be the graph obtained by removing exactly one edge from K4 and let k be an integer with k ≥ 2. We prove that if G is a claw-free graph of order at least 13k - 12 and with minimum degree at least five, then G contains k vertex-disjoint copies of K4. The requirement of number five is necessary.展开更多
A graph is said to be K1,4-free if it does not contain an induced subgraph isomorphic to K1,4. Let κ be an integer with κ ≥ 2. We prove that if G is a K1,4-free graph of order at least llκ- 10 with minimum degree ...A graph is said to be K1,4-free if it does not contain an induced subgraph isomorphic to K1,4. Let κ be an integer with κ ≥ 2. We prove that if G is a K1,4-free graph of order at least llκ- 10 with minimum degree at least four, then G contains k vertex-disjoint copies of K1 ＋ （K1 ∪ KK2）.展开更多
Let G be a graph, and k a positive integer. A graph G is fractional independent-set-deletable k-factor-critical(in short, fractional ID-k-factor-critical) if G-I has a fractional k-factor for every independent set I o...Let G be a graph, and k a positive integer. A graph G is fractional independent-set-deletable k-factor-critical(in short, fractional ID-k-factor-critical) if G-I has a fractional k-factor for every independent set I of G. In this paper, we present a sufficient condition for a graph to be fractional ID-k-factor-critical,depending on the minimum degree and the neighborhoods of independent sets. Furthermore, it is shown that this result in this paper is best possible in some sense.展开更多
文摘In this paper the authors generalize the classic random bipartite graph model, and define a model of the random bipartite multigraphs as follows：let m = m（n） be a positive integer-valued function on n and ζ（n,m;{pk}） the probability space consisting of all the labeled bipartite multigraphs with two vertex sets A ={a_1,a_2,...,a_n} and B = {b_1,b_2,...,b_m}, in which the numbers t_（ai）,b_j of the edges between any two vertices a_i∈A and b_j∈ B are identically distributed independent random variables with distribution P{t_（ai）,b_j=k}=pk,k=0,1,2,...,where pk ≥0 and ∞Σk=0 pk=1. They obtain that X_（c,d,A）, the number of vertices in A with degree between c and d of G_（n,m）∈ζ（n, m;{pk}） has asymptotically Poisson distribution, and answer the following two questions about the space ζ（n,m;{pk}） with {pk} having geometric distribution, binomial distribution and Poisson distribution, respectively. Under which condition for {pk} can there be a function D（n） such that almost every random multigraph G_（n,m）∈ζ（n,m;{pk}） has maximum degree D（n）in A？ under which condition for {pk} has almost every multigraph G（n,m）∈ζ（n,m;{pk}） a unique vertex of maximum degree in A？
基金supported by the National Natural Science Foundation of China(Nos.11371052,11731002)the Fundamental Research Funds for the Central Universities(Nos.2016JBM071,2016JBZ012)the 111 Project of China(B16002)
文摘Let a, b, r be nonnegative integers with 1 ≤ a ≤ b and r ≥ 2. Let G be a graph of order n with n 〉（a＋2 b）（r（a＋b）-2）/b.In this paper, we prove that G is fractional ID-[a, b]-factor-critical if δ（G）≥bn/a＋2 b＋a（r-1）and ｜NG（x1） ∪ NG（x2） ∪…∪ NG（xr）｜ ≥（a＋b）n/（a＋2 b） for any independent subset {x1,x2,…,xr} in G. It is a generalization of Zhou et al.＇s previous result [Discussiones Mathematicae Graph Theory, 36： 409-418（2016）]in which r = 2 is discussed. Furthermore, we show that this result is best possible in some sense.
文摘The harmonic index of a graph?G? is defined as where d(u) denotes the degree of a vertex u in G . In this work, we give another expression for the Harmonic index. Using this expression, we give the minimum value of the harmonic index for any triangle-free graphs with order n and minimum degree δ ≥ k for k≤ n/2? and show the corresponding extremal graph is the complete graph.
基金Supported in part by the National Natural Science Foundation of China under Grant No.11401102,11271307 and 11101086Fuzhou university of Science and Technology Development Fund No.2014-XQ-29
文摘In this paper we consider the random r-uniform r-partite hypergraph model H（n1, n2,…, nr; n, p） which consists of all the r-uniform r-partite hypergraphs with vertex partition {V1,V2,…, Vr} where ｜Vi｜ = ni = ni（n） （1 ≤ i ≤ r） are positive integer-valued functions on n with n1 ＋n2 ＋… ＋nr =n, and each r-subset containing exactly one element in Vi （1 ≤ i ≤ r） is chosen to be a hyperedge of Hp ∈H（n1,n2,…,nr;n,p） with probability p = p（n）, all choices being independent. Let △V1 = △V1 （H） and δv1 = δv1（H） be the maximum and minimum degree of vertices in V1 of H, respectively; Xd,V1 = Xd,V1 （H）, Yd,V1 = Yd,V1 （H）, Zd,V1 = Zd,V1 （H） and Zc,d,V1=Zc,d,V1 （H） be the number of vertices in V1 of H with degree d, at least d, at most d, and between c and d, respectively. In this paper we obtain that in the space H（n1, n2,…, nr; n,p）, Xd,V1, Yd,V1, Zd,V1, and Zc,d,V1all have asymptotically Poisson distributions. We also answer the following two questions. What is the range of p that there exists a function D（n） such that in the space H（n1, n2,…,nr; n, p）, lim n→＋∞ P（△v1 = D（n）） = 1？ What is the range of p such that a.e., Hp ∈ H（n1,n2,...,nr;n,p） has a unique vertex in V1 with degree Av1（Hp）？ Both answers are p = o（logn1/N）, where N = r ∏ i=2 ni. The corresponding problems on δv1（Hp） also are considered, and we obtained the answers are p ≤ （1＋o（1））（logn1/N） andp=o （log n1/N）, respectively.
基金Supported in part by two grants from Ministerio de Economía y Competitividad,Spain:MTM2013-46374-P and MTM2015-69323-REDT
文摘In this work, we obtain good upper bounds for the diameter of any graph in terms of its minimum degree and its order, improving a classical theorem due to Erdos, Pach, Pollack and Tuza. We use these bounds in order to study hyperbolic graphs （in the Gromov sense）. To compute the hyperbolicity constant is an almost intractable problem, thus it is natural to try to bound it in terms of some parameters of the graph. Let H（n, δ0） be the set of graphs G with n vertices and minimum degree 50, and J（n, Δ） be the set of graphs G with n vertices and maximum degree A. We study the four following extremal problems on graphs： a（n,δ0） = min{δ（G） ｜ G ∈H（n, δ0）}, b（n, δ0） =- max{δ（G）｜ e ∈H（n, δ0）}, α（n, Δ） = min{δ（G） [ G ∈ J（n, Δ）} and β（n,Δ） = max{δ（G） ] G∈Π（n,Δ）}. In particular, we obtain bounds for b（n, δ0） and we compute the precise value of a（n, δ0）, α（n, Δ） and w（n, Δ） for all values of n, r0 and A, respectively.
基金Supported by two grants from Ministerio de Economía y Competitividad,Spain(Grant Nos.MTM2013-46374-P and MTM2015-69323-REDT)
文摘The original version of the article was published in [1]. Unfortunately, the original version of this article contains a mistake： in Theorem 6.2 appears that β（n, △） = （n-△ ＋ 5）/4 but the correct statement is β（n, △） = （n -△ ＋ 4）/4. In this erratum we correct the theorem and give the correct proof.
基金This work was supported in part by the National Natural Science Foundation of China （Grant Nos. 11161035, 11401455） and the Fundamental Research Funds for the Central Universities （No. K5051370010）.
文摘A graph is said to be claw-free if it does not contain an induced subgraph isomorphic to K1,3. Let K4 be the graph obtained by removing exactly one edge from K4 and let k be an integer with k ≥ 2. We prove that if G is a claw-free graph of order at least 13k - 12 and with minimum degree at least five, then G contains k vertex-disjoint copies of K4. The requirement of number five is necessary.
基金Supported by National Natural Science Foundation of China(Grant Nos.11161035 and 11226292)Ningxia Ziran(Grant No.NZ1153)research grant from Ningxia University(Grant No.zr1122)
文摘A graph is said to be K1,4-free if it does not contain an induced subgraph isomorphic to K1,4. Let κ be an integer with κ ≥ 2. We prove that if G is a K1,4-free graph of order at least llκ- 10 with minimum degree at least four, then G contains k vertex-disjoint copies of K1 ＋ （K1 ∪ KK2）.
基金supported by the National Natural Science Foundation of China(Grant No.11371009,11501256,61503160)Six Big Talent Peak of Jiangsu Province(Grant No.JY–022)+3 种基金333 Project of Jiangsu Provincethe National Social Science Foundation of China(Grant No.14AGL001)the Natural Science Foundation of Xinjiang Province of China(Grant No.2015211A003)the Natural Science Foundation of the Higher Education Institutions of Jiangsu Province(Grant No.14KJD110002)
文摘Let G be a graph, and k a positive integer. A graph G is fractional independent-set-deletable k-factor-critical(in short, fractional ID-k-factor-critical) if G-I has a fractional k-factor for every independent set I of G. In this paper, we present a sufficient condition for a graph to be fractional ID-k-factor-critical,depending on the minimum degree and the neighborhoods of independent sets. Furthermore, it is shown that this result in this paper is best possible in some sense.
