A signed k-partite graph (signed multipartite graph) is a k-partite graph in which each edge is assigned a positive or a negative sign. If G(V1, V2, · · · , Vk) is a signed k-partite graph with Vi = {vi1, vi2, · · · , vini}, 1 ≤ i ≤ k, the signed degree of vij is sdeg(vij) = dij = d + ij − d − ij , where 1 ≤ i ≤ k, 1 ≤ j ≤ ni and d + ij(d − ij) is the number of positive (negative) edges inciden...

Tur\'an's Theorem says that an extremal $K_{r+1}$-free graph is $r$-partite. The Stability of Erd\H{o}s and Simonovits shows if a with $n$ vertices has close to the maximal $t_r(n)$ edges, then it being In this paper we determine exactly graphs at least $m$ edges are farthest from $r$-partite, for any $m\ge t_r(n) - \delta_r n^2$. This extends work by Erd\H{o}s, Gy\H{o}ri Simonovits, proves con...

We prove that for all r ≥ 2 and c > 0, every G graph of order n with at least cnr cliques of order r contains a complete r-partite graph with each part of size ⌊cr log n⌋ . This result implies a concise form of the Erdős-Stone theorem.

Let Fn,tr(n) denote the family of all graphs on n vertices and tr(n) edges, where tr(n) is the number of edges in the Turán’s graph Tr(n) – the complete r-partite graph on n vertices with partition sizes as equal as possible. For a graph G and a positive integer λ, let PG(λ) denote the number of proper vertex colorings of G with at most λ colors, and let f(n, tr(n), λ) = max{PG(λ) : G ∈ Fn,tr(n...

Let D be a digraph, V (D) and A(D) will denote the sets of vertices and arcs of D, respectively. A (k, l)-kernel N of D is a k-independent set of vertices (if u, v ∈ N then d(u, v), d(v, u) ≥ k) and l-absorbent (if u ∈ V (D) − N then there exists v ∈ N such that d(u, v) ≤ l). A k-kernel is a (k, k − 1)-kernel. A digraph D is cyclically k-partite if there exists a partition {Vi} i=0 of V (D) suc...

A tournament is an orientation of a complete graph, and in general a multipartite or c-partite tournament is an orientation of a complete c-partite graph. In a recent article, the authors proved that a regular c-partite tournament with r ≥ 2 vertices in each partite set contains a cycle with exactly r− 1 vertices from each partite set, with exception of the case that c = 4 and r = 2. Here we wi...

We put the final piece into a puzzle first introduced by Bollobás, Erdős and Szemerédi in 1975. For arbitrary positive integers n and r we determine the largest integer ∆ = ∆(r, n), for which any r-partite graph with partite sets of size n and of maximum degree less than ∆ has an independent transversal. This value was known for all even r. Here we determine the value for odd r and find that ∆(...

Let D(U1, U2, ..., Uk) be an oriented k -partite graph with |Ui| = ni, 1 ≤ i ≤ k. Then, the score of a vertex ui in Ui is defined by aui (or simply ai ) = ∑k t=1, t 6=i nt+d + ui −d − ui , where d+ui and d − ui are respectively the outdegree and indegree of ui. The set A of distinct scores of the vertices of D(U1, U2, ..., Uk) is called its score set. In this paper, we prove that if a1 is a non...

The distance matrix of a simple graph G is D(G) = (di,j), where di,j is the distance between the ith and jth vertices of G. The greatest eigenvalue λ1 of D(G) is called the distance spectral radius of the graph G and is denoted by λ1(G). A simple connected graph G is called a 2-partite distance regular graph if there exists a partition V1 ∪V2 of the vertex set of G such that for i = 1, 2 and an...

It is proven that for every fixed h, a and b, a graph with n vertices and minimum degree at least h−1 h n, which contains no copy of Kb (the complete graph with b vertices), contains at least (1 − o(1)) n ha vertex disjoint induced copies of the complete h-partite graph with a vertices in each color class. Mathematics subject classification numbers: 05C70, 05C55. All graphs considered here are ...