For all integers k with k≥2, if G is a balanced k-partite graph on n≥3 vertices minimum degree at least⌈n2⌉+⌊n+22⌈k+12⌉⌋−nk={⌈n2⌉+⌊n+2k+1⌋−nk:k odd n2+⌊n+2k+2⌋−nk:k even , then has Hamiltonian cycle unless 4 divides n and k∈{2,n2}. In the case where k∈{2,n2}, we can characterize graphs which do not have see that ⌈n2⌉+⌊n+22⌈k+12⌉⌋−nk+1 suffices. This result tight for k≥2 divisible by k.

The notion of a graph theoretic Ramsey number is generalised by assuming that both the original graph whose edges are arbitrarily bi–coloured and the sought after monochromatic subgraphs are complete, balanced, multipartite graphs, instead of complete graphs as in the classical definition. We previously confined our attention to diagonal multipartite Ramsey numbers. In this paper the definition...

A k-cycle decomposition of a multipartite graph G is said to be gregarious if each k-cycle in the decomposition intersects k distinct partite sets of G. In this paper we prove necessary and sufficient conditions for the existence of such a decomposition in the case where G is the complete equipartite graph, having n parts of size m, and either n ≡ 0, 1 (mod k), or k is odd and m ≡ 0 (mod k). As...

In this paper, we present an adaptive graph-based personalized recommendation method based on co-ranking and query-based collaborative diffusion. By utilizing the unique network structure of n-partite heterogeneous graph, we attempt to address the problem of personalized recommendation in a two-layer ranking process with the help of reasonable measure of high and low order relationships and ana...

Given a graph G and a natural number k, the k graph product of G = (V,E) is the graph with vertex set V . For every two vertices x = (x1, . . . , xk) and y = (y1, . . . , yk) in V , an edge is placed according to a predefined rule. Graph products are a basic combinatorial object, widely studied and used in different areas such as hardness of approximation, information theory, etc. We study grap...

A sequence of nonnegative integers π is graphic if it is the degree sequence of some graph G. In this case we say that G is a realization of π, and we write π = π(G). A graphic sequence π is potentially H-graphic if there is a realization of π that contains H as a subgraph. Given nonincreasing graphic sequences π1 = (d1, . . . , dn) and π2 = (s1, . . . , sn), we say that π1 majorizes π2 if di ≥...

Let q be a positive integer, and G be a q-partite simple graph on qn vertices, with n vertices in each vertex class. Let δ = k k+1 + λ, where k = q + 4dlog qe and λ is a positive real number. If each vertex of G is adjacent to at least δn vertices in each of the other vertex classes, q is bounded and n is large enough, then G has a Kq-factor.

For a given graph H let φH (n) be the maximum number of parts that are needed to partition the edge set of any graph on n vertices such that every member of the partition is either a single edge or it is isomorphic to H. Pikhurko and Sousa conjectured that φH (n) = ex(n,H) for χ(H) ≥ 3 and all sufficiently large n, where ex(n,H) denotes the maximum size of a graph on n vertices not containing H...

Suppose a graph G have n vertices, m edges, and t triangles. Letting λn (G) be the largest eigenvalue of the Laplacian of G and μn (G) be the smallest eigenvalue of its adjacency matrix, we prove that λn (G) ≥ 2m2 − 3nt m (n2 − 2m) , μn (G) ≤ 3n3t− 4m3 nm (n2 − 2m) , with equality if and only if G is a regular complete multipartite graph. Moreover, if G is Kr+1-free, then λn (G) ≥ 2mn (r − 1) (...

A digraph obtained by replacing each edge of a complete m-partite graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a semicomplete m-partite digraph. We describe results (theorems and algorithms) on directed walks in semicomplete mpartite digraphs including some recent results concerning tournaments.