The paper studies the domatic numbers and the total domatic numbers of graphs having cut-vertices. We shall study the domatic number d(G) and the total domatic number d t (G) of a graph G. A survey of the related theory is given in 3]. We consider nite, undirected graphs without loops or multiple edges. A subset D of the vertex set V (G) of a graph G is called dominating (total dominating), if ...

Given a graph G, we say that a subset D of the vertex set V is a dominating set if it is near all the vertices, in that every vertex outside of D is adjacent to a vertex in D. A domatic k-partition of G is a partition of V into k dominating sets. In this paper, we will consider issues of computability related to domatic partitions of computable graphs. Our investigation will center on answering...

A k-tuple coloring of a graph G assigns a set of k colors to each vertex of G such that if two vertices are adjacent, the corresponding sets of colors are disjoint. The k-tuple chromatic number of G, χk(G), is the smallest t so that there is a k-tuple coloring of G using t colors. It is well known that χ(G2H) = max{χ(G), χ(H)}. In this paper, we show that there exist graphs G and H such that χk...

Let G be a connected graph. Let f be a proper k -coloring of G and Π = (R_1, R_2, . . . , R_k) bean ordered partition of V (G) into color classes. For any vertex v of G, define the color code c_Π(v) of v with respect to Π to be a k -tuple (d(v, R_1), d(v, R_2), . . . , d(v, R_k)), where d(v, R_i) is the min{d(v, x)|x ∈ R_i}. If distinct vertices have distinct color codes, then we call f a locat...

A k-tuple coloring of a graph G assigns a set of k colors to each vertex of G so that if two vertices are adjacent, the corresponding sets of colors are disjoint. The k-tuple chromatic number of G is the smallest t so that there is such a k-tuple coloring of G using t colors in all. The Kneser graph K(m,n) has as vertices all n-element subsets of the set {1,2, . . . ,m} and an edge between two ...