Given a collection of matchings M = (M1,M2, . . . ,Mq) (repetitions allowed), a matching M contained in ⋃ M is said to be s-rainbow for M if it contains representatives from s matchings Mi (where each edge is allowed to represent just one Mi). Formally, this means that there is a function φ : M → [q] such that e ∈ Mφ(e) for all e ∈ M , and |Im(φ)| > s. Let f(r, s, t) be the maximal k for which ...

Generally, a graph G, an independent set is a subset S of vertices in G such that no two vertices in S are adjacent (connected by an edge) and a vertex cover is a subset S of vertices such that each edge of G has at least one of its endpoints in S. Again, the minimum vertex cover problem is to find a vertex cover with the smallest number of vertices. Consider a k-partite graph G = (V, E) with v...

A graph G is called a complete k-partite (k ≥ 2) graph if its vertices can be partitioned into k independent sets V1, . . . , Vk such that each vertex in Vi is adjacent to all the other vertices in Vj for 1 ≤ i < j ≤ k. A complete k-partite graph G is a complete balanced kpartite graph if |V1| = |V2| = · · · = |Vk|. An edge-coloring of a graph G with colors 1, . . . , t is an interval t-colorin...

A (finite or infinite) complete bipartite graph together with some end vertices all adjacent to a common vertex is called a complete bipartite graph with a horn. For any bipartite graph G, we show that G is the graph of a commutative semigroup with 0 if and only if it is one of the following graphs: star graph, two-star graph, complete bipartite graph, complete bipartite graph with a horn. We a...

Let H be an r-partite r-graph, all of whose sides have the same size n. Suppose that there exist two sides of H , each satisfying the following condition: the degree of each legal r−1-tuple contained in the complement of this side is strictly larger than n 2 . We prove that under this condition H must have a perfect matching. This answers a question of Kühn and Osthus.

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...

LetKn1,n2,...,np denote the complete p-partite graph, p > 1, on n = n1+n2+ · · ·+np vertices and let n1 ≥ n2 ≥ · · · ≥ np > 0. We show that for a fixed value of n, both the spectral radius and the energy of complete p-partite graphs are minimal for complete split graph CS(n, p− 1) and are maximal for Turán graph T (n, p).

For integers k > 0 and 0 < r ≤ ∆ (where r ≤ k), a conditional (k, r)-coloring of a graph G is a proper k-coloring of the vertices of G such that every vertex v of degree d(v) in G is adjacent to vertices with at least min{r, d(v)} differently colored neighbors. The smallest integer k for which a graph G has a conditional (k, r)-coloring is called the rth order conditional chromatic number, deno...

For n > r > 1, let f,(n) denote the minimum number q, such that it is possible to partition all edges of the complete r-graph on n vertices into q complete r-partite r-graphs. Graham and Pollak showed that fz(n) = n-1. Here we observe that f3(n) = n-2 and show that for every fixed r > 2, there are positive constants cx(r) and c2(r) such that q(r) < f,(n)" n-f'/2J < c2(r) for all n > r. This sol...