This paper provides lower orientable k-step domination number and upper orientable k-step domination number of complete r-partite graph for 1 ≤ k ≤ 2. It also proves that the intermediate value theorem holds for the complete r-partite graphs.

graph is called a complete r-partite graph if its vertex set can be partitioned into r independent sets V1, . . . , Vr such that for any i = 1, 2, . . . , r every vertex in Vi is adjacent to all other vertices in Vj , j 6= i. We denote a complete r-partite graph with part sizes |Vi| = ni by Kn1,...,nr . For a graph G and a positive integer k we use kG to denote k vertex-disjoint copies of G. Gi...

A graph G is called integral if all the eigenvalues of its adjacency matrix are integers. In this paper we investigate integral complete r−partite graphs Kp1,p2,...,pr = Ka1p1,a2p2,...,asps with s ≤ 4. New sufficient conditions for complete 3-partite graphs and complete 4-partite graphs to be integral are given. From these conditions we construct infinitely many new classes of integral complete...

We strengthen the stability theorem of Erdős and Simonovits. Write Kr (s1, . . . , sr) for the complete r-partite graph with parts of size s1, . . . , sr and Tr (n) for the r-partite Turán graph of order n. Our main result is: For all r ≥ 2 and sufficiently small c > 0, ε > 0, for every graph G of order n with ⌈ (1− 1/r − ε)n2/2 ⌉ edges, one of the following conditions holds: (a) Kr+1 ( ⌊c lnn⌋...

The trace norm ‖G‖∗ of a graph G is the sum of its singular values, i.e., the absolute values of its eigenvalues. The norm ‖G‖∗ has been intensively studied under the name of graph energy, a concept introduced by Gutman in 1978. This note focuses on the maximum trace norm of r-partite graphs, which for r > 2 raises some unusual problems. It is shown that, if G is an r-partite graph of order n, ...

Let K+ r (s1, . . . , sr) be the complete r-partite graph with parts of size s1 ≥ 2, s2, . . . , sr with an edge added to the first part. Letting tr (n) be the number of edges of the r-partite Turán graph of order n, we prove that: (A) For all r ≥ 2 and all sufficiently small ε > 0, every graph of sufficiently large order n with tr (n) + 1 edges contains a K + r ( ⌊c lnn⌋ , . . . , ⌊c lnn⌋ , ⌈ ...

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 ...

The choice number of a graph G is the smallest integer k such that for every assignment of a list L(v) of k colors to each vertex v of G, there is a proper coloring of G that assigns to each vertex v a color from L(v). We present upper and lower bounds on the choice number of complete multipartite graphs with partite classes of equal sizes and complete r-partite graphs with r−1 partite classes ...

We consider two extremal problems in hypergraphs. First, given k ≥ 3 and k-partite k-uniform hypergraphs, as a generalization of graph (k = 2) matchings, we determine the partite minimum codegree threshold for matchings with at most one vertex left in each part, thereby answering a problem asked by Rödl and Ruciński. We further improve the partite minimum codegree conditions to sum of all k par...