An automorphism group G of a 1-factorization of the complete multipartite graph Km×n consists of permutations of the vertices of the graph mapping factors to factors. In this paper, we give a complete answer to the existence problem of a 1-factorization of Km×n admitting a cyclic or dihedral group acting sharply transitively on the vertices of the graph.

A semicomplete multipartite digraph is obtained by replacing each edge of a complete multipartite graph by an arc or by a pair of two mutually opposite arcs. Very recently, Yeo 7] proved that every regular semicomplete multipartite digraph is Hamiltonian. With this, Yeo connrmed a conjecture of C.-Q. Zhang 8]. In the rst part of this paper, a generalization of regularity is considered. We exten...

A digraph obtained by replacing each edge of a complete multipartite graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a semicomplete multipartite digraph. L. Volkmann (1998) raised the following question: Let D be a strong semicomplete multipartite digraph with a longest path of length l. Does there exist a strong spanning oriented subgraph of D with a lo...

In this paper, we first establish a useful inequality on the minimum real roots of the adjoint polynomials of the complete graphs. By using it, we investigate the chromatic uniqueness of certain complete multipartite graphs. An unsolved problem (i.e., Problem 11), posed by Koh and Teo in Graph and Combin. 6(1990) 259–285, is completely solved by giving it a positive answer. Moreover, many exist...

A graphG is called chromatic-choosable if its choice number is equal to its chromatic number, namely Ch(G) = χ(G). Ohba has conjectured that every graph G satisfying |V (G)| ≤ 2χ(G)+1 is chromatic-choosable. Since each k-chromatic graph is a subgraph of a complete k-partite graph, we see that Ohba’s conjecture is true if and only if it is true for every complete multipartite graph. However, the...

An intense activity is nowadays devoted to the definition of models capturing the properties of complex networks. Among the most promising approaches, it has been proposed to model these graphs via their clique incidence bipartite graphs. However, this approach has, until now, severe limitations resulting from its incapacity to reproduce a key property of this object: the overlapping nature of ...

For each vertex v of a graph G, if there exists a list of k colors, L(v), such that there is a unique proper coloring for G from this collection of lists, then G is called a uniquely k-list colorable graph. Ghebleh and Mahmoodian characterized uniquely 3-list colorable complete multipartite graphs except for nine graphs: K2,2,r r ∈ {4, 5, 6, 7, 8}, K2,3,4, K1∗4,4, K1∗4,5, K1∗5,4. Also, they con...

We exactly evaluate the entanglement of a six vertex and a nine vertex graph states which correspond to non ”twocolorable” graphs. The upper bound of entanglement for five vertices ring graph state is improved to 2.9275, less than upper bound determined by LOCC. An upper bound of entanglement is proposed based on the definition of graph state. PACS number(s): 03.67.Mn, 03.65.Ud, Keyword(s):grap...