Competitively orientable complete multipartite graphs

Integral complete multipartite graphs

A graph is called integral if all eigenvalues of its adjacency matrix are integers. In this paper, we investigate integral complete r-partite graphsKp1,p2,...,pr =Ka1·p1,a2·p2,...,as ·ps with s=3, 4.We can construct infinite many new classes of such integral graphs by solving some certain Diophantine equations. These results are different from those in the existing literature. For s = 4, we giv...

Complete subgraphs in multipartite graphs

Turán’s Theorem states that every graphG of edge density ‖G‖/ (|G| 2 ) > k−2 k−1 contains a complete graph K and describes the unique extremal graphs. We give a similar Theorem for `-partite graphs. For large `, we find the minimal edge density d` , such that every `-partite graph whose parts have pairwise edge density greater than d` contains a K . It turns out that d` = k−2 k−1 for large enou...

Decompositions of complete multipartite graphs

This paper answers a recent question of Dobson and Marušič by partitioning the edge set of a complete bipartite graph into two parts, both of which are edge sets of arctransitive graphs, one primitive and the other imprimitive. The first member of the infinite family is the one constructed by Dobson and Marušič.

On graphs with complete multipartite µ-graphs

Homogeneous factorisations of complete multipartite graphs

A homogeneous factorisation of a graph is a partition of its arc set such that there exist vertex transitive subgroups M < G 6 Aut(Γ) with M fixing each part of the partition setwise and G preserving the partition and transitively permuting the parts. In this paper we study homogeneous factorisations of complete multipartite graphs such that M acts regularly on vertices. We provide a necessary ...