On 5‐Cycles and 6‐Cycles in Regular n‐Tournaments

Let T be a tournament of order n and be the number of cycles of length m in T. For and odd n, the maximum of is achieved for any regular tournament of order n (M. G. Kendall and B. Babington Smith, 1940), and in the case it is attained only for the unique regular locally transitive tournament RLT_n of order n (U. Colombo, 1964). A lower bound was also obtained for in the class of regular tournaments of order n (A. Kotzig, 1968). This bound is attained if and only if T is doubly regular (when ) or nearly doubly regular (when ) (B. Alspach and C. Tabib, 1982). In the present article, we show that for any regular tournament T of order n, the equality holds. This allows us to reduce the case to the case In turn, the pure spectral expression for obtained in the class implies that for a regular tournament T of order the inequality holds, with equality if and only if T is doubly regular or T is the unique regular tournament of order 7 that is neither doubly regular nor locally transitive. We also determine the value of c₆(RLT_n) and conjecture that this value coincides with the minimum number of 6‐cycles in the class for each odd      

传递的二部竞赛图的性质

Abipartitetournamentistransitiveifitcontainsnocycles.Becausetransitivem ×nbi partitetournamentisn’tuniqueintheisomorphicsense,theproblemsoftransitivebipartitetournamentsarecomplicated .Hence ,BEINEKELWandMOONJW gaveacriterionforde terminingwhetherascoreorderedpaircontainssometransitivebipartitetournament(see [1 ]) .Theenumerationoftransitivebipartitetournamentswasdiscussedby [2 ].LetTm ,n =(V ,U ,E) beam ×nbipartitetournamentwith｜V｜=mand｜U｜=n .Denotebyd(w)thescoreofvertexw .Th…     

Non‐Critical Vertices and Long Circuits in Strong Tournaments of Order n and Diameter d

Let T be a strong tournament of order with diameter . A vertex w in T is non‐critical if the subtournament is also strong. In the opposite case, it is a critical vertex. In the present article, we show that T has at most critical vertices. This fact and Moon's vertex‐pancyclic theorem imply that for , it contains at least circuits of length . We describe the class of all strong tournaments of order with diameter for which this lower bound is achieved and show that for , the minimum number of circuits of length in a tournament of this class is equal to . In turn, the minimum among all strong tournaments of order with diameter 2 grows exponentially with respect to n for any given . Finally, for , we select a strong tournament of order n with diameter d and conjecture that for any strong tournament T of order n whose diameter does not exceed d, the number of circuits of length ? in T is not less than that in for each possible ?. Moreover, if these two numbers are equal to each other for some given , where , then T is isomorphic to either or its converse . For , this conjecture was proved by Las Vergnas. In the present article, we confirm it for the case . In an Appendix, some problems concerning non‐critical vertices are considered for generalizations of tournaments.     

Romanoff定理的定量形式

1934年,Romanoff证明了能表成2的方幂与一个素数之和形式的正整数在正整数集合中有正的比例．最近,本文作者证明了对充分大的x,能表成2的方幂与一个素数之和形式的正整数在不超过x的正整数中至少有0．0868x个．本文证明了:设 x≥5,则在不超过x的正整数中,能表成2的方幂与一个素数之和的数的个数不少于 0．005x,即给出了Romanoff定理的定量形式．     

BCI-代数的一个结构定理

证明了下面的结构定理：每一个非零的周期BCI-代数是一些次直不可约BCI-代数的次直积．     

没有线性结构的极大极小定理与鞍点定理

在没有线性结构的一般集合上引入了函数的一种“凹（凸）”性概念，得到一个没有线性结构的ＦａｎＫｙ不等式；在此基础上，在一般的拓扑空间上建立了极大极小定理，并把著名的鞍点定理推广到没有线性结构的拓扑空间上·     

Landau's inequalities for tournament scores and a short proof of a theorem on transitive sub‐tournaments

Ao and Hanson, and Guiduli, Gyárfás, Thomassé and Weidl independently, proved the following result: For any tournament score sequence S = (s₁, s₂, … ,s_n) with s₁≤s₂ ≤ … ≤ s_n, there exists a tournament T on vertex set {1,2, …, n} such that the score of each vertex i is s_i and the sub‐tournaments of T on both the even and the odd indexed vertices are transitive in the given order; that is, i dominates j whenever i > j and i ≡ j (mod 2). In this note, we give a much shorter proof of the result. In the course of doing so, we show that the score sequence of a tournament satisfies a set of inequalities which are individually stronger than the well‐known set of inequalities of Landau, but collectively the two sets of inequalities are equivalent. © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 244–254, 2001     

No 17‐player triplewhist tournament has nontrivial automorphisms

The existence of triplewhist tournaments for v players has recently been solved for all values of v except = 17. For v = 12 and v = 13, a complete enumeration has shown the nonexistence of TWh(v), while constructions of TWh(v) have been presented for v > 17. For several values of v, existence has been shown by constructing a TWh(v) with a prescribed, usually cyclic, automorphism group. In this article, it is shown that the strategy of constructing a TWh(v) with a prescribed automorphism group cannot succeed with TWh(17), since no 17‐player triplewhist tournament has nontrivial automorphisms. © 2004 Wiley Periodicals, Inc.     

Finite 2‐Geodesic Transitive Graphs of Prime Valency

We classify noncomplete prime valency graphs satisfying the property that their automorphism group is transitive on both the set of arcs and the set of 2‐geodesics. We prove that either Γ is 2‐arc transitive or the valency p satisfies , and for each such prime there is a unique graph with this property: it is a nonbipartite antipodal double cover of the complete graph with automorphism group and diameter 3.     

Disjoint 3‐Cycles in Tournaments: A Proof of The Bermond–Thomassen Conjecture for Tournaments

We prove that every tournament with minimum out‐degree at least contains k disjoint 3‐cycles. This provides additional support for the conjecture by Bermond and Thomassen that every digraph D of minimum out‐degree contains k vertex disjoint cycles. We also prove that for every , when k is large enough, every tournament with minimum out‐degree at least contains k disjoint cycles. The linear factor 1.5 is best possible as shown by the regular tournaments.     

Bombieri's Theorem in Short Intervals

Under the assumption of sixth power large sieve mean-value of Dirichlet L-function,we improve Bombieri's theorem in short intervals by virtue of the large sieve method and Heath-Brown's identity.     

关于抽象素数定理的形式(英文)

In this paper we prove a zero-free region for L-functions LG（z,Х）. As an application, an abstract prime number theorem with sharp error-term for formations is established.     

Flag‐Transitive Point‐Primitive Symmetric (ν,κ,λ) Designs With λ at Most 100

Let be a symmetric (ν,κ,λ) design with λ ≤ 100. If G is a flag‐transitive and point‐primitive automorphism group of , then G must be an affine or almost simple group.     

学生化U-统计量的中心极限定理

在β（１＜β３／２）阶矩存在的条件下，我们证明了学生化Ｕ 统计量服从中心极限定理．     

On Finite 2‐Path‐Transitive Graphs

The class of graphs that are 2‐path‐transitive but not 2‐arc‐transitive is investigated. The amalgams for such graphs are determined, and structural information regarding the full automorphism groups is given. It is then proved that a graph is 2‐path‐transitive but not 2‐arc‐transitive if and only if its line graph is half‐arc‐transitive, thus providing a method for constructing new families of half‐arc‐transitive graphs. © 2012 Wiley Periodicals, Inc. J. Graph Theory 73: 225–237, 2013     

On the prime geodesic theorem for hyperbolic 3‐manifolds

Through the Selberg zeta approach, we reduce the exponent in the error term of the prime geodesic theorem for cocompact Kleinian groups or Bianchi groups from Sarnak's to . At the cost of excluding a set of finite logarithmic measure, the bound is further improved to .     

Doubly transitive 2‐factorizations

Let be a 2‐factorization of the complete graph K_v admitting an automorphism group G acting doubly transitively on the set of vertices. The vertex‐set V(K_v) can then be identified with the point‐set of AG(n, p) and each 2‐factor of is the union of p‐cycles which are obtained from a parallel class of lines of AG(n, p) in a suitable manner, the group G being a subgroup of A G L(n, p) in this case. The proof relies on the classification of 2‐(v, k, 1) designs admitting a doubly transitive automorphism group. The same conclusion holds even if G is only assumed to act doubly homogeneously. © 2006 Wiley Periodicals, Inc. J Combin Designs     

3‐Factor‐Criticality of Vertex‐Transitive Graphs

A graph of order n is p ‐factor‐critical, where p is an integer of the same parity as n, if the removal of any set of p vertices results in a graph with a perfect matching. 1‐factor‐critical graphs and 2‐factor‐critical graphs are factor‐critical graphs and bicritical graphs, respectively. It is well known that every connected vertex‐transitive graph of odd order is factor‐critical and every connected nonbipartite vertex‐transitive graph of even order is bicritical. In this article, we show that a simple connected vertex‐transitive graph of odd order at least five is 3‐factor‐critical if and only if it is not a cycle.