A digraph obtained by replacing each edge of a complete mpartite graph with an arc or a pair of mutually opposite arcs with the same end vertices is called a complete m-partite digraph. We describe an O(n) algorithm for finding a longest path in a complete m-partite (m ≥ 2) digraph with n vertices. The algorithm requires time O(n) in case of testing only the existence of a Hamiltonian path and ...

the chromatic number of a graph g, denoted by χ(g), is the minimum number of colors such that g can be colored with these colors in such a way that no two adjacent vertices have the same color. a clique in a graph is a set of mutually adjacent vertices. the maximum size of a clique in a graph g is called the clique number of g. the turán graph tn(k) is a complete k-partite graph whose partition...

An outerplanar (also called circular, convex, one-page) drawing of an n-vertex graph G is a drawing in which the vertices are placed on a circle and each edge is drawn using one straight-line segment. We derive exact results for the minimal number of crossings in any outerplanar drawings of the following classes of graphs: 3-row meshes, Halin graphs and complete p−partite graphs with equal size...

For graphs H and F, the generalized Turán number ex(n,H,F) is largest of copies in an F-free graph on n vertices. We say that F-Turán-good if (χ(F)−1)-partite graph, provided large enough. present a general theorem case F has edge whose deletion decreases chromatic number. In particular, this determines ex(n,Pk,C2ℓ+1) ex(n,C2k,C2ℓ+1) exactly, also study when vertex

We prove three results about the spectral radius μ (G) of a graph G : (a) Let Tr (n) be the r-partite Turán graph of order n. If G is a Kr+1-free graph of order n, then μ (G) < μ (Tr (n)) unless G = Tr (n) . (b) For most irregular graphs G of order n and size m, μ (G)− 2m/n > 1/ (2m+ 2n) . (c) Let 0 ≤ k ≤ l. If G is a graph of order n with no K2 +Kk+1 and no K2,l+1, then μ (G) ≤ min {

For r 4 we determine the smallest number of vertices, .<71-( d), of complete that are decomposable into two isomorphic factors for a given finite diameter d. We also prove that for a ,d such graph exists for each order than gr( d). 1. INTRODUCTORY NOTES AND DEFINITIONS In this paper we study decompositions of finite complete multipartite graphs factors with prescribed diameter. A factor F of a ...

Let Sk(n) be the maximum number of orientations an n-vertex graph G in which no copy Kk is strongly connected. For all integers n, k ≥ 4 where n 5 or 5, we prove that = 2tk - 1(n), tk-1(n) edges (k 1)-partite Turán Tk-1(n). Moreover, Tk-1(n) only having 2tk-1(n) with connected copies Kk.

Modern recommender systems (RS) work by processing a number of signals that can be inferred from large sets user-item interaction data. The main signal to analyze stems the raw matrix represents interactions. However, we increase performance RS considering other kinds like context interactions, which could be, for example, time or date interaction, user location, sequential data corresponding h...

Let A be a non-trivial, finitely-generated abelian group and A∗ = A\{0}. A graph is A-magic if there exists an edge labeling using elements of A∗ which induces a constant vertex labeling of the graph. In this paper, we analyze the group-magic property for complete n-partite graphs and composition graphs with deleted edges.

A nonincreasing sequence of nonnegative integers π = (d1, d2, ..., dn) is graphic if there is a (simple) graph G of order n having degree sequence π. In this case, G is said to realize π. For a given graph H, a graphic sequence π is potentially H-graphic if there is some realization of π containing H as a (weak) subgraph. Let σ(π) denote the sum of the terms of π. For a graph H and n ∈ Z+, σ(H,...