On k‐ary spanning trees of tournaments

It is well known that every tournament contains a Hamiltonian path, which can be restated as that every tournament contains a unary spanning tree. The purpose of this article is to study the general problem of whether a tournament contains a k‐ary spanning tree. In particular, we prove that, for any fixed positive integer k, there exists a minimum number h(k) such that every tournament of order at least h(k) contains a k‐ary spanning tree. The existence of a Hamiltonian path for any tournament is the same as h(1) = 1. We then show that h(2) = 4 and h(3) = 8. The values of h(k) remain unknown for k ≥ 4. © 1999 John & Sons, Inc. J Graph Theory 30: 167–176, 1999     

Arc-disjoint hamiltonian paths in non-round decomposable local tournaments

Thomassen proved that a strong tournament $T$ has a pair of arc-disjoint Hamiltonian paths with distinct initial vertices and distinct terminal vertices if and only if $T$ is not an almost transitive tournament of odd order, where an almost transitive tournament is obtained from a transitive tournament with acyclic ordering $u_1,u_2,\dots ,u_n$ (i.e., $u_i\to u_j$ for all $1\le i<j\le n$) by reversing the arc $u_1{u}_n$. A digraph $D$ is a local tournament if for every vertex $x$ of $D$, both the out-neighbors and the in-neighbors of $x$ induce tournaments. Bang-Jensen, Guo, Gutin and Volkmann split local tournaments into three subclasses: the round decomposable; the non-round decomposable which are not tournaments; the non-round decomposable which are tournaments. In 2015, we proved that every 2-strong round decomposable local tournament has a Hamiltonian path and a Hamiltonian cycle which are arc-disjoint if and only if it is not the second power of an even cycle. In this paper, we discuss the arc-disjoint Hamiltonian paths in non-round decomposable local tournaments, and prove that every 2-strong non-round decomposable local tournament contains a pair of arc-disjoint Hamiltonian paths with distinct initial vertices and distinct terminal vertices. This result combining with the one on round decomposable local tournaments extends the above-mentioned result of Thomassen to 2-strong local tournaments.     

On arc-traceable local tournaments

A digraph D is arc-traceable if for every arc xy of D, the arc xy belongs to a directed Hamiltonian path of D. A local tournament is an oriented graph such that the negative neighborhood as well as the positive neighborhood of every vertex induces a tournament. It is well known that every tournament contains a directed Hamiltonian path and, in 1990, Bang-Jensen showed the same for connected local tournaments. In 2006, Busch, Jacobson and Reid studied the structure of tournaments that are not arc-traceable and consequently gave various sufficient conditions for tournaments to be arc-traceable. Inspired by the article of Busch, Jacobson and Reid, we develop in this paper the structure necessary for a local tournament to be not arc-traceable. Using this structure, we give sufficient conditions for a local tournament to be arc-traceable and we present examples showing that these conditions are best possible.     

哈密尔顿多部竞赛图

本文证明了如下结果：设Ｔ为几乎正则ｎ－部竞赛图（ｎ　７）,则Ｔ必含哈密尔顿圈．     

图的能量与哈密尔顿性

设G是一个无向简单图,A（G）为G的邻接矩阵．用G的补图的特征值给出G包含哈密尔顿路、哈密尔顿圈以及哈密尔顿连通图的充分条件：其次用二部图的拟补图的特征值给出二部图包含哈密尔顿圈的充分条件．这些结果改进了一些已知的结果．     

图的能量与哈密尔顿性

设G是一个无向简单图, A(G)为$G$的邻接矩阵. 用G的补图的特征值给出G包含哈密尔顿路、哈密尔顿圈以及哈密尔顿连通图的充分条件; 其次用二部图的拟补图的特征值给出二部图包含哈密尔顿圈的充分条件. 这些结果改进了一些已知的结果.     

Improved approximation algorithm for the feedback set problem in a bipartite tournament

We present a simple 3-approximation algorithm for the feedback vertex set problem in a bipartite tournament, improving on the approximation ratio of 3.5 achieved by the best previous algorithms.     

二部竞赛图中的AD路与AD回路

本文证明了：若对二部竞赛图Ｔ的每一顶点ｖ，总有ｍｉｎ｛ｄＴ＾＋（ｖ），ｄＴ＾－（ｖ）｝≥ｋ≥３，则Ｔ中存在长度至少为４ｒ的ＡＤ路或ＡＤ回路，除非Ｔ同构于一类例外图之一。作为推论，我们得到：正则二部竞赛图Ｔ含有ＡＤＨ回路，除非Ｔ属于一类例外图。     

HAMILTONIAN DECOMPOSITION OF COMPLETE BIPARTITE γ-HYPERGRAPHS

1. IntroductionHypergraphs are most general structure in discrete mathematics. Computer scientistsintroduced the concept of acyclic hypergraphs and proved that acyc1ic hypergraphs are veryuseful in database theory['--']. In [l], a new system of axioms fOr paths, connectivity andcycles of hypergraphs was introduced. In this paper, we wiIl introduce the concepts ofbipartite hypergraphs and Hamiltonian paths and cycles of a hypergraph.Let V be a finite set. n = (V e) is called a hypergraph on …     

Antidirected Hamiltonian paths and directed cycles in tournaments

We prove that in a tournament of odd order $n\ge 5$, the number of antidirected Hamiltonian paths starting with a forward arc and the number of Hamiltonian circuits have the same parity.     

Hamiltonian decomposition of complete bipartiter-hypergraphs

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 withq vertices in each part is Hamiltonian decomposable whereq is a prime. This research is supported by the National Natural Science Foundation of China (No.19831080).     

推点与二部竞赛图的强连通性

设D是一个有向图,S是V(D)的子集．在D中推S,是指颠倒D中所有的只有一个端点在S中的弧的方向． Klostermeyer提出了对于任给的一个有向图D,能否通过推点使之成为强连通的有向图的问题．他证明了上述判定问题是NP-完备的．而我们论证了对于任意的二部竞赛图D,如果V(D)的二划分是(X,Y),并满足3≤|X|≤|Y|≤2|X|-1-1, 则可以通过推点使D成为强连通的有向图,而且,|Y|的上界2|X|-1-1是最好可能的．     

Signed domination numbers of directed graphs

The concept of signed domination number of an undirected graph (introduced by J. E. Dunbar, S. T. Hedetniemi, M. A. Henning and P. J. Slater) is transferred to directed graphs. Exact values are found for particular types of tournaments. It is proved that for digraphs with a directed Hamiltonian cycle the signed domination number may be arbitrarily small.     

Hamiltonian cycles in bipartite graphs

We give a sufficient condition for bipartite graphs to be Hamiltonian. The condition involves the edge-density and balanced independence number of a bipartite graph.     

Edge‐disjoint Hamiltonian cycles in hypertournaments

We introduce a method for reducing k‐tournament problems, for k ≥ 3, to ordinary tournaments, that is, 2‐tournaments. It is applied to show that a k‐tournament on n ≥ k + 1 + 24d vertices (when k ≥ 4) or on n ≥ 30d + 2 vertices (when k = 3) has d edge‐disjoint Hamiltonian cycles if and only if it is d‐edge‐connected. Ironically, this is proved by ordinary tournament arguments although it only holds for k ≥ 3. We also characterizatize the pancyclic k‐tournaments, a problem posed by Gutin and Yeo.(Our characterization is slightly incomplete in that we prove it only for n large compared to k.). © 2005 Wiley Periodicals, Inc. J Graph Theory     

The cycle structure of regular multipartite tournaments

A multipartite tournament is an orientation of a complete multipartite graph. A tournament is a multipartite tournament, each partite set of which contains exactly one vertex. Alspach (Canad. Math. Bull. 10 (1967) 283) proved that every regular tournament is arc-pancyclic. Although all partite sets of a regular multipartite tournament have the same cardinality, Alspach's theorem is not valid for regular multipartite tournaments. In this paper, we prove that if the cardinality common to all partite sets of a regular n-partite (n3) tournament T is odd, then every arc of T is in a cycle that contains vertices from exactly m partite sets for all m{3,4,…,n}. This result extends Alspach's theorem for regular tournaments to regular multipartite tournaments. We also examine the structure of cycles through arcs in regular multipartite tournaments.     

Every cycle-connected multipartite tournament has a universal arc

A digraph D is cycle-connected if for every pair of vertices u,v∈V(D) there exists a directed cycle in D containing both u and v. In 1999, Ádám [A. Ádám, On some cyclic connectivity properties of directed graphs, Acta Cybernet. 14 (1) (1999) 1-12] posed the following problem. Let D be a cycle-connected digraph. Does there exist a universal arc in D, i.e., an arc e∈A(D) such that for every vertex w∈V(D) there is a directed cycle in D containing both e and w?A c-partite or multipartite tournament is an orientation of a complete c-partite graph. Recently, Hubenko [A. Hubenko, On a cyclic connectivity property of directed graphs, Discrete Math. 308 (2008) 1018-1024] proved that each cycle-connected bipartite tournament has a universal arc. As an extension of this result, we show in this note that each cycle-connected multipartite tournament has a universal arc.     

Hamiltonian-connected tournaments

We characterize weakly Hamiltonian-connected tournaments and weakly panconnected tournaments completely and we apply these results to cycles and bypasses in tournaments with given irregularity, in particular, in regular and almost regular tournaments. We give a sufficient condition in terms of local and global connectivity for a Hamiltonian path with prescribed initial and terminal vertex. From this result we deduce that every 4-connected tournament is strongly Hamiltonian-connected and that every edge of a 3-connected tournament is contained in a Hamiltonian cycle of the tournament and we describe infinite families of tournaments demonstrating that these results are best possible.     

Hamilton cycles,avoiding prescribed arcs,in close-to-regular tournaments

The local irregularity of a digraph D is defined as i_l(D) = max {|d⁺ (x) − d⁻ (x)| : x ϵ V(D)}. Let T be a tournament, let Γ = {V₁, V₂, …, V_c} be a partition of V(T) such that |V₁| ≥ |V₂| ≥ … ≥ |V_c|, and let D be the multipartite tournament obtained by deleting all the arcs with both end points in the same set in Γ. We prove that, if |V(T)| ≥ max{2i_l(T) + 2|V₁| + 2|V₂| − 2, i_l(T) + 3|V₁| − 1}, then D is Hamiltonian. Furthermore, if T is regular (i.e., i_l(T) = 0), then we state slightly better lower bounds for |V(T)| such that we still can guarantee that D is Hamiltonian. Finally, we show that our results are best possible. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 123–136, 1999