The case k = 3 of a complete k-partite graph is called tripartite Tp,q,r. It that its vertices are decomposed into three disjoint sets such that, no two within the same set adjacent. has recently attracted much attention due to important in several applications, especially, chemistry where some molecular orbital compounds correspondents structure. One method capturing structure through computin...

Given a graph property P, it is interesting to determine the typical structure of graphs that satisfy P. In this paper, we consider monotone properties, that is, properties that are closed under taking subgraphs. Using results from the theory of graph limits, we show that if P is a monotone property and r is the largest integer for which every r-colorable graph satis es P, then almost every gra...

The p-spectral radius of a graph G of order n is defined for any real number p > 1 as λ (G) = max 2 ∑ {i,j}∈E(G) xixj : x1, . . . , xn ∈ R and |x1| + · · ·+ |xn| = 1  . The most remarkable feature of λ(p) is that it seamlessly joins several other graph parameters, e.g., λ(1) is the Lagrangian, λ(2) is the spectral radius and λ(∞)/2 is the number of edges. This paper presents solutions to so...

We shall assume that the reader is familiar with standard terminology on directed graphs (see, e.g., Bang-Jensen and Gutin [1]). In this note, if we speak of a cycle, then we mean a directed cycle. If xy is an arc of a digraph D, then we write x → y and say x dominates y. If X and Y are two disjoint vertex sets of a digraph D such that every vertex of X dominates every vertex of Y , then we say...

Let K` denote the complete graph on ` vertices. We prove that there is a constant c = c(`), such that whenever p ≥ n−c, with probability tending to 1 when n goes to infinity, every maximum K`-free subgraph of the binomial random graph Gn,p is (`− 1)partite. This answers a question of Babai, Simonovits and Spencer [BSS90]. The proof is based on a tool of independent interest: we show, for instan...

The set S of distinct scores (outdegrees) of the vertices of a k-partite tournament T(X1, X2, · · · , X k) is called its score set. In this paper, we prove that every set of n non-negative integers, except {0} and {0, 1}, is a score set of some 3-partite tournament. We also prove that every set of n non-negative integers is a score set of some k-partite tournament for every n ≥ k ≥ 2.

The concept of the incidence chromatic number of a graph was introduced by Brualdi and Massey (1993). They conjectured that every graph G can be incidence colored with ∆(G) + 2 colors. In this paper, we calculate the incidence chromatic numbers of the complete k-partite graphs and give the incidence chromatic number of three infinite families of graphs.

In this note, we show that a complete k-partite graph is the only graph with clique number k among all degree-equivalent simple graphs. This result gives a lower bound on the clique number, which is sharper than existing bounds on a large family of graphs.

The concept of the incidence chromatic number of a graph was introduced by Brualdi and Massey (1993). They conjectured that every graph G can be incidence colored with ∆(G) + 2 colors. In this paper, we calculate the incidence chromatic numbers of the complete k-partite graphs and give the incidence chromatic number of three infinite families of graphs.