The complete multipartite graph with d partite sets, each containing m vertices, will be denoted by K(d; m). If all of the partite sets are not of the same cardinality, the graph K(mi, m2, . . . . md) is the complete d-partite graph with m, vertices in each partite sets, 1 <i<d. The special case where mi= 1 for every i, yields the complete graph on d vertices and is denoted by Kd. The join of g...

It is known that for a graph on n vertices bn2/4c + 1 edges is sufficient for the existence of many triangles. In this paper, we determine the minimum number of edges sufficient for the existence of k triangles intersecting in exactly one common vertex. 1 Notation With integers n ≥ p ≥ 1, we let Tn,p denote the Turán graph, i.e., the complete p-partite graph on n vertices where each partite set...

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. An m-partite tournament is an orientation of an m-partite complete graph. In this pap...

Given a graph H, we say a graph G is H-saturated if G does not contain H as a subgraph and the addition of any edge e′ 6∈ E(G) results in H as a subgraph. In this paper, we construct (K4 − e)-saturated graphs with |E(G)| either the size of a complete bipartite graph, a 3-partite graph, or in the interval [ 2n− 4, ⌊ n 2 ⌋ ⌈ n 2 ⌉ − n+ 6 ] . We then extend the (K4−e)-saturated graphs to (Kt − e)-...

In the following material, we use the notion of an independent set of a graph G: this is a set of I vertices of G such that no two vertices of I are adjacent in G. These are sometimes referred to as stable sets in the literature. The chromatic number of a graph G, denoted χ(G), is the smallest r such that V (G) has a partition into independent sets V1, V2, . . . , Vr. In other words, we may ass...

We address the following problem: Given a complete k-partite geometric graph K whose vertex set is a set of n points in R, compute a spanner of K that has a “small” stretch factor and “few” edges. We present two algorithms for this problem. The first algorithm computes a (5 + )-spanner of K with O(n) edges in O(n log n) time. The second algorithm computes a (3 + )-spanner of K with O(n log n) e...

Let F be a fixed graph of chromatic number r + 1. We prove that for all large n the degree sequence of any F -free graph of order n is, in a sense, close to being dominated by the degree sequence of some r-partite graph. We present two different proofs: one goes via the Regularity Lemma and the other uses a more direct counting argument. Although the latter proof is longer, it gives better esti...

We give some new bounds for the clique and independence numbers of a graph in terms of its eigenvalues. In particular we prove the following results. Let G be a graph of order n, average degree d, independence number α (G) , and clique number ω (G) . (i) If μn is the smallest eigenvalue of G, then ω (G) ≥ 1 + dn (n− d) (d− μn) . Equality holds if and only if G is a complete regular ω-partite gr...

The tree partition number of an r-edge-colored graph G, denoted by tr(G), is the minimum number k such that whenever the edges of G are colored with r colors, the vertices of G can be covered by at most k vertex-disjoint monochromatic trees. We determine t2(K(n1, n2, . . . , nk)) of the complete k-partite graph K(n1, n2, . . . , nk). In particular, we prove that t2(K(n,m)) = (m − 2)/2n + 2, whe...

For a set of graphs F, let ex(n,F) and spex(n,F) denote the maximum number edges spectral radius an n-vertex F-free graph, respectively. Nikiforov (LAA, 2007) gave version Turán Theorem by showing that spex(n,Kk+1)=λ(Tk(n)), where Tk(n) is k-partite graph on n vertices. In same year, Feng, Yu Zhang (LAA) determined exact value spex(n,Ms+1), Ms+1 matching with s+1 edges. Recently, Alon Frankl (a...