Yubao Guo

Cycles in multipartite tournaments

An n-partite tournament is an orientation of a complete n-partite graph. We show that if D is a strongly connected n-partite (n ≥ 3) tournament, then every partite set of D has at least one vertex, which lies on an m-cycle for all m in {3, 4,..., n}. This result extends those of Bondy (J. London Math. Soc.14 (1976), 277-282) and Gutin (J. Combin. Theory Ser. B58 (1993), 319-321).

Connectivity properties of locally semicomplete digraphs

It is shown that every k-connected locally semicomplete digraph D with minimum outdegree at least 2k and minimum indegree at least 2k − 2 has at least m = max{2, k} vertices x1, x2, , xm such that D − xi is k-connected for i = 1, 2, , m.

The 1-good-neighbour diagnosability of Cayley graphs generated by transposition trees under the PMC model and MM* model

ABSTRACT Diagnosability is an important metric for measuring the reliability of multiprocessor systems. In 2012, Peng et al. proposed a new measure for fault tolerance of the system, which is called g-good-neighbour diagnosability that restrains every fault-free node containing at least g fault-free neighbours. As a favourable topology structure of interconnection networks, the Cayley graph generated by the transposition tree has many good properties. In this paper, we give that the 1-good-neighbour diagnosability of under the PMC model and MM model is except the bubble-sort graph under MM model, where , and the 1-good-neighbour diagnosability of under the MM model is 4.

On complementary cycles in locally semicomplete digraphs

Abstract In this paper we show that a 2-connected locally semicomplete digraph of order at least 8 is not cycle complementary if and only if it is 2-diregular and has odd order. This result yields immediately two conjectures of Bang-Jensen.

A note on vertex pancyclic oriented graphs

It was conjectured in [Wang, to appear in The Australasian Journal of Combinatorics] that, for each integer k ≥ 2, there exists M(k) such that if G= (V1, V2, E) is a bipartite graph with |V1| = |V2| = n ≥ M(k) and d(x) + d(y) ≥ n + k for each pair of nonadjacent vertices x and y of G with x e V1 and y e V2, then, for any k independent edges e1, …, ek of G, there exist k vertex-disjoint cycles C1, …, Ck in G such that ei e E(Ci for all i e {1, …, k} and V(C1 ⊎ ··· ∪ Ck) = V(G). This conjecture is also verified for k = 2, 3 in [Wang, to appear; Wang, manuscript]. In this article, we prove this conjecture to be true if n ≥ 3k, i.e., M(k) ≤ 3k. We will also show that, if n ≥ 3k, then for any k independent edges e1, …, ek of G, there exist k vertex-disjoint cycles C1, …, Ck of length at most 6 in G such that ei e E(Ci) for all i e {1, …, k}.

On cycles through a given vertex in multipartite tournaments

Abstract An n -partite tournament is an orientation of a complete n -partite graph, and an m -cycle is a directed cycle of length m . If D is a strongly connected n -partite tournament with the partite sets V 1 , V 2 , …, V n and v an arbitrary vertex of D , then we shall prove the following statements. • • The vertex v is contained in a longest cycle. • • If n ≥ 3, then v belongs to an m -cycle or ( m + 1)-cycle for all m ϵ {3,4, …, n }. • • If n ⩾ 4 and | Vi | ⩾ 2 for i = 1,2, …, n , then v belongs to an ( n + 1)-cycle or ( n + 2)-cycle. As easy consequences, we obtain known results of Bondy (1976), Ayel (1981), and Gutin (1984).

Pancyclic out-arcs of a vertex in tournaments

Abstract Thomassen (J. Combin. Theory Ser. B 28, 1980 , 142–163) proved that every strong tournament contains a vertex x such that each arc going out from x is contained in a Hamiltonian cycle. In this paper, we extend the result of Thomassen and prove that a strong tournament contains a vertex x such that every arc going out from x is pancyclic, and our proof yields a polynomial algorithm to find such a vertex. Furthermore, as another consequence of our main theorem, we get a result of Alspach (Canad. Math. Bull. 10, 1967 , 283–286) that states that every arc of a regular tournament is pancyclic.

A New Sufficient Condition for a Digraph to be Hamiltonian

In \cite{suffcond} the following extension of Meyniels theorem was conjectured: If

Path-connectivity in local tournaments

D

Degree conditions on distance 2 vertices that imply k-ordered Hamiltonian

is a digraph on