In this paper we present some results connected with still open problem of Gauss, negative Pell’s equation and some type graphs.In particular we prove in the Theorem 1 that all real quadratic fields K=Q( ) , generate...In this paper we present some results connected with still open problem of Gauss, negative Pell’s equation and some type graphs.In particular we prove in the Theorem 1 that all real quadratic fields K=Q( ) , generated by Fermat’s numbers with d=Fm+1=22m+1+1,m≥2, have not unique factorization. Theorem 2 give a connection of the Gauss problem with primitive Pythagorean triples. Moreover, in final part of our paper we indicate on some connections of the Gauss problem with odd graphs investigated by Cremona and Odoni in the papper [5].展开更多
The following results are obtained： （1） The graph Cm^m· Pt is odd arithmetic when （i） m ≡ 0 （mod 2） and t=m or m ＋ 1; （ii） m ≡ 1 （mod 2） and t=m ＋ 1. （2） The graph C2m^m is odd arithmetic when （...The following results are obtained： （1） The graph Cm^m· Pt is odd arithmetic when （i） m ≡ 0 （mod 2） and t=m or m ＋ 1; （ii） m ≡ 1 （mod 2） and t=m ＋ 1. （2） The graph C2m^m is odd arithmetic when （i） m=2,4 and n is any positive integer; （ii） m=3 and n is even. （3） The graph Cm^m, is odd arithmetic when m=4n and t=2. （4） Pm＋1^n is odd arithmetic when （i） n is odd; （ii） m 〈 3 and n is any positive integer. （5） Windmill graph Kn^t is odd arithmetic if and only if n=2. （6） Cycle Cn is odd arithmetic if and only if n ≡ 0 （mod 4）. （7） For any positive integer n and any positive integer m, Km,n is odd arithmetic.展开更多
A k-fold n-coloring of G is a mapping φ: V (G) → Zk(n) where Zk(n) is the collection of all ksubsets of {1,2,...,n} such that φ(u) ∩φ(v) = φ if uv ∈ E(G).If G has a k-fold n-coloring,i.e.,G is k-fold n-colorabl...A k-fold n-coloring of G is a mapping φ: V (G) → Zk(n) where Zk(n) is the collection of all ksubsets of {1,2,...,n} such that φ(u) ∩φ(v) = φ if uv ∈ E(G).If G has a k-fold n-coloring,i.e.,G is k-fold n-colorable.Let the smallest integer n such that G is k-fold n-colorable be the k-th chromatic number,denoted by χk(G).In this paper,we show that any outerplanar graph is k-fold 2k-colorable or k-fold χk(C*)-colorable,where C* is a shortest odd cycle of G.Moreover,we investigate that every planar graph with odd girth at least 10k-9(k 3) can be k-fold (2k + 1)-colorable.展开更多
Let G be a graph. If there exists a spanning subgraph F such that dF(x) ∈ {1,3,…2n – 1}, then is called to be (1,2n – 1)-odd factor of G. Some sufficient and necessary conditions are given for G – U to have (1,2n...Let G be a graph. If there exists a spanning subgraph F such that dF(x) ∈ {1,3,…2n – 1}, then is called to be (1,2n – 1)-odd factor of G. Some sufficient and necessary conditions are given for G – U to have (1,2n – 1)-odd factor where U is any subset of V(G) such that |U| = k.展开更多
We determine all square-free odd positive integers n such that the 2-Selmer groups Sn and (S)n of the elliptic curve En: y2 = x(x - n)(x - 2n) and its dual curve (E)n: y2 =x3 + 6nx2 + n2x have the smallest size: Sn = ...We determine all square-free odd positive integers n such that the 2-Selmer groups Sn and (S)n of the elliptic curve En: y2 = x(x - n)(x - 2n) and its dual curve (E)n: y2 =x3 + 6nx2 + n2x have the smallest size: Sn = {1}, (S)n = {1, 2, n, 2n}. It is well known that for such integer n, the rank of group En(Q) of the rational points on En is zero so that n is a non-congruent number. In this way we obtain many new series of elliptic curves En with rank zero and such series of integers n are non-congruent numbers.展开更多
文摘In this paper we present some results connected with still open problem of Gauss, negative Pell’s equation and some type graphs.In particular we prove in the Theorem 1 that all real quadratic fields K=Q( ) , generated by Fermat’s numbers with d=Fm+1=22m+1+1,m≥2, have not unique factorization. Theorem 2 give a connection of the Gauss problem with primitive Pythagorean triples. Moreover, in final part of our paper we indicate on some connections of the Gauss problem with odd graphs investigated by Cremona and Odoni in the papper [5].
基金the Natural Science Foundation of Hebei Province and Mathematical Center （No. 08M002）.
文摘The following results are obtained： （1） The graph Cm^m· Pt is odd arithmetic when （i） m ≡ 0 （mod 2） and t=m or m ＋ 1; （ii） m ≡ 1 （mod 2） and t=m ＋ 1. （2） The graph C2m^m is odd arithmetic when （i） m=2,4 and n is any positive integer; （ii） m=3 and n is even. （3） The graph Cm^m, is odd arithmetic when m=4n and t=2. （4） Pm＋1^n is odd arithmetic when （i） n is odd; （ii） m 〈 3 and n is any positive integer. （5） Windmill graph Kn^t is odd arithmetic if and only if n=2. （6） Cycle Cn is odd arithmetic if and only if n ≡ 0 （mod 4）. （7） For any positive integer n and any positive integer m, Km,n is odd arithmetic.
基金supported by National Natural Science Foundation of China (Grant No.10971198)the Natural Science Foundation of Zhejiang Province,China (Grant No.Y607467)
文摘A k-fold n-coloring of G is a mapping φ: V (G) → Zk(n) where Zk(n) is the collection of all ksubsets of {1,2,...,n} such that φ(u) ∩φ(v) = φ if uv ∈ E(G).If G has a k-fold n-coloring,i.e.,G is k-fold n-colorable.Let the smallest integer n such that G is k-fold n-colorable be the k-th chromatic number,denoted by χk(G).In this paper,we show that any outerplanar graph is k-fold 2k-colorable or k-fold χk(C*)-colorable,where C* is a shortest odd cycle of G.Moreover,we investigate that every planar graph with odd girth at least 10k-9(k 3) can be k-fold (2k + 1)-colorable.
文摘Let G be a graph. If there exists a spanning subgraph F such that dF(x) ∈ {1,3,…2n – 1}, then is called to be (1,2n – 1)-odd factor of G. Some sufficient and necessary conditions are given for G – U to have (1,2n – 1)-odd factor where U is any subset of V(G) such that |U| = k.
基金This work was supported by the National Scientific Research Project 973 of China(Grant No.2004 CB 3180004)the National Natural Science Foundation of China(Grant No.60433050).
文摘We determine all square-free odd positive integers n such that the 2-Selmer groups Sn and (S)n of the elliptic curve En: y2 = x(x - n)(x - 2n) and its dual curve (E)n: y2 =x3 + 6nx2 + n2x have the smallest size: Sn = {1}, (S)n = {1, 2, n, 2n}. It is well known that for such integer n, the rank of group En(Q) of the rational points on En is zero so that n is a non-congruent number. In this way we obtain many new series of elliptic curves En with rank zero and such series of integers n are non-congruent numbers.
