A mathematical algorithm of the distribution of greenhouse gas emissions is proposed as a way to tackle the topical issue of climate change and develop approaches to attaining an agreement among emitters of greenhouse...A mathematical algorithm of the distribution of greenhouse gas emissions is proposed as a way to tackle the topical issue of climate change and develop approaches to attaining an agreement among emitters of greenhouse gases (on the global scale, in a country, a region, a megalopolis).展开更多
We single out the polygonal paths of nodd -1 order that solve each of the different longest non-cyclic Euclidean Hamiltonian path problems in networks by an arithmetic algorithm. As by product, the procedure determine...We single out the polygonal paths of nodd -1 order that solve each of the different longest non-cyclic Euclidean Hamiltonian path problems in networks by an arithmetic algorithm. As by product, the procedure determines the winding index of cyclic Hamiltonian polygonals on the vertices of a regular polygon.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
The paper explores the connection of Graph-Lagrangians and its maximum cliques for 3-uniform hypergraphs. Motzkin and Straus showed that the Graph-Lagrangian of a graph is the Graph-Lagrangian of its maximum cliques. ...The paper explores the connection of Graph-Lagrangians and its maximum cliques for 3-uniform hypergraphs. Motzkin and Straus showed that the Graph-Lagrangian of a graph is the Graph-Lagrangian of its maximum cliques. This connection provided a new proof of Turin classical result on the Turan density of complete graphs. Since then, Graph-Lagrangian has become a useful tool in extremal problems for hypergraphs. Peng and Zhao attempted to explore the relationship between the Graph-Lagrangian of a hypergraph and the order of its maximum cliques for hypergraphs when the number of edges is in certain range. They showed that if G is a 3-uniform graph with m edges containing a clique of order t - 1, then A（G） = A（[t- 1]（3）） provided （t31） ≤ m ≤ （3^t1） ＋ （2^rt-2）. They also conjectured： If G is an r-uniform graph with m edges not containing a clique of order t - 1, then A（G） 〈 A（[t - 1]（r）） provided （r^t-1） ≤ m ≤ （r^t-1） ＋ （r-1^t-2）. It has been shown that to verify this conjecture for 3-uniform graphs, it is sufficient to verify the conjecture for left-compressed 3-uniform graphs with m = （3^t-1） ＋ （2^t-2）. Regarding this conjecture, we show： If G is a left-compressed 3-uniform graph on the vertex set It] with m edges and lit - 1]（3） ／ E（G）｜=- p, then A（G） 〈 A（[t - 1]（3）） provided m = （3^t-1） ＋ （2^t-2） and t ≥ 17p/2 ＋ 11.展开更多
Let be an undirected graph. The maximum cycle packing problem in G then is to find a collection of edge-disjoint cycles C<sub>i</sup>in G such that s is maximum. In general, the maximum cycle packing probl...Let be an undirected graph. The maximum cycle packing problem in G then is to find a collection of edge-disjoint cycles C<sub>i</sup>in G such that s is maximum. In general, the maximum cycle packing problem is NP-hard. In this paper, it is shown for even graphs that if such a collection satisfies the condition that it minimizes the quantityon the set of all edge-disjoint cycle collections, then it is a maximum cycle packing. The paper shows that the determination of such a packing can be solved by a dynamic programming approach. For its solution, an-shortest path procedure on an appropriate acyclic networkis presented. It uses a particular monotonous node potential.展开更多
Frankl and Füredi in [1] conjectured that the r-graph with m edges formed by taking the first m sets in the colex ordering of N^（r） has the largest Lagrangian of all r-graphs with m edges.Denote this r-graph by...Frankl and Füredi in [1] conjectured that the r-graph with m edges formed by taking the first m sets in the colex ordering of N^（r） has the largest Lagrangian of all r-graphs with m edges.Denote this r-graph by Cr,m and the Lagrangian of a hypergraph by λ（G）.In this paper,we first show that if（3^t-1） ≤m〈（3^t）,G is a left-compressed 3-graph with m edges and on vertex set[t],the triple with minimum colex ordering in G^c is（t — 2 — i）（t — 2）t,then λ（G） ≤λ（C3,m）.As an implication,the conjecture of Frankl and Fiiredi is true for（3^t）-6≤m≤（3^t）.展开更多
Some extremal properties of the integral of Legendre polynomials are given, which are of independent interest. Meanwhile they show that a conjecture of P. Erdos[1] is plausible and maybe provides some means to prove t...Some extremal properties of the integral of Legendre polynomials are given, which are of independent interest. Meanwhile they show that a conjecture of P. Erdos[1] is plausible and maybe provides some means to prove this conjecture.展开更多
文摘A mathematical algorithm of the distribution of greenhouse gas emissions is proposed as a way to tackle the topical issue of climate change and develop approaches to attaining an agreement among emitters of greenhouse gases (on the global scale, in a country, a region, a megalopolis).
文摘We single out the polygonal paths of nodd -1 order that solve each of the different longest non-cyclic Euclidean Hamiltonian path problems in networks by an arithmetic algorithm. As by product, the procedure determines the winding index of cyclic Hamiltonian polygonals on the vertices of a regular polygon.
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Supported in part by National Natural Science Foundation of China(Grant No.11271116)
文摘The paper explores the connection of Graph-Lagrangians and its maximum cliques for 3-uniform hypergraphs. Motzkin and Straus showed that the Graph-Lagrangian of a graph is the Graph-Lagrangian of its maximum cliques. This connection provided a new proof of Turin classical result on the Turan density of complete graphs. Since then, Graph-Lagrangian has become a useful tool in extremal problems for hypergraphs. Peng and Zhao attempted to explore the relationship between the Graph-Lagrangian of a hypergraph and the order of its maximum cliques for hypergraphs when the number of edges is in certain range. They showed that if G is a 3-uniform graph with m edges containing a clique of order t - 1, then A（G） = A（[t- 1]（3）） provided （t31） ≤ m ≤ （3^t1） ＋ （2^rt-2）. They also conjectured： If G is an r-uniform graph with m edges not containing a clique of order t - 1, then A（G） 〈 A（[t - 1]（r）） provided （r^t-1） ≤ m ≤ （r^t-1） ＋ （r-1^t-2）. It has been shown that to verify this conjecture for 3-uniform graphs, it is sufficient to verify the conjecture for left-compressed 3-uniform graphs with m = （3^t-1） ＋ （2^t-2）. Regarding this conjecture, we show： If G is a left-compressed 3-uniform graph on the vertex set It] with m edges and lit - 1]（3） ／ E（G）｜=- p, then A（G） 〈 A（[t - 1]（3）） provided m = （3^t-1） ＋ （2^t-2） and t ≥ 17p/2 ＋ 11.
文摘Let be an undirected graph. The maximum cycle packing problem in G then is to find a collection of edge-disjoint cycles C<sub>i</sup>in G such that s is maximum. In general, the maximum cycle packing problem is NP-hard. In this paper, it is shown for even graphs that if such a collection satisfies the condition that it minimizes the quantityon the set of all edge-disjoint cycle collections, then it is a maximum cycle packing. The paper shows that the determination of such a packing can be solved by a dynamic programming approach. For its solution, an-shortest path procedure on an appropriate acyclic networkis presented. It uses a particular monotonous node potential.
基金Supported by Chinese Universities Scientic Fund(No.N140504004)the National Natural Science Foundation of China(No.11271116)
文摘Frankl and Füredi in [1] conjectured that the r-graph with m edges formed by taking the first m sets in the colex ordering of N^（r） has the largest Lagrangian of all r-graphs with m edges.Denote this r-graph by Cr,m and the Lagrangian of a hypergraph by λ（G）.In this paper,we first show that if（3^t-1） ≤m〈（3^t）,G is a left-compressed 3-graph with m edges and on vertex set[t],the triple with minimum colex ordering in G^c is（t — 2 — i）（t — 2）t,then λ（G） ≤λ（C3,m）.As an implication,the conjecture of Frankl and Fiiredi is true for（3^t）-6≤m≤（3^t）.
基金the National Natural Science Foundation of China (No.19671082).
文摘Some extremal properties of the integral of Legendre polynomials are given, which are of independent interest. Meanwhile they show that a conjecture of P. Erdos[1] is plausible and maybe provides some means to prove this conjecture.
