On the Delta(d)–chromatic number of a complete balanced multipartite graph

The interval number of a complete multipartite graph

The interval number of a graph G, denoted i(G), is the least positive integer t for which G is the intersection graph of a family of sets each of which is the union of at most t cIosed intervals of the real line IR. Trotter and Harary showed that the interval number of the complete bipartite graph K(m, n) is [(mn + I)/(m + n)]. Matthews showed that the interval number of the complete multiparti...

Heterochromatic tree partition number of a complete multipartite graph

A heterochromatic tree is an edge-colored tree in which any two edges have different colors. The heterochromatic tree partition number of an r-edge-colored graph G, denoted by tr(G), is the minimum k such that whenever the edges of the graph G are colored with r colors, the vertices of G can be covered by at most k vertexdisjoint heterochromatic trees. In this paper we determine the heterochrom...

Interval colorings of complete balanced multipartite graphs

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...

Crossing numbers of complete tripartite and balanced complete multipartite graphs

The crossing number cr(G) of a graph G is the minimum number of crossings in a nondegenerate planar drawing of G. The rectilinear crossing number cr(G) of G is the minimum number of crossings in a rectilinear nondegenerate planar drawing (with edges as straight line segments) of G. Zarankiewicz proved in 1952 that cr(Kn1,n2) ≤ Z(n1, n2) := ⌊ n1 2 ⌋ ⌊ n1−1 2 ⌋ ⌊ n2 2 ⌋ ⌊ n2−1 2 ⌋ . We define an ...

The diameter of an orientation of a complete multipartite graph