Besides the classical chromatic and achromatic numbers of a graph related to minimum or minimal vertex partitions into independent sets, the b-chromatic number was introduced in 1998 thanks to an alternative definition of the minimality of such partitions. When independent sets are replaced by dominating sets, the parameters corresponding to the chromatic and achromatic numbers are the domatic ...

This paper surveys some of the work that was inspired by Wagner’s general technique to prove completeness in the levels of the boolean hierarchy over NP and some related results. In particular, we show that it is DP-complete to decide whether or not a given graph can be colored with exactly four colors, where DP is the second level of the boolean hierarchy. This result solves a question raised ...

An extremely simple, linear time algorithm is given for constructing a domatic partition in totally balanced hypergraphs. This simpli es and generalizes previous algorithms for interval and strongly chordal graphs. On the other hand, the domatic number problem is shown to be NP-complete for several families of perfect graphs, including chordal and bipartite graphs.

An {em Italian dominating function} on a digraph $D$ with vertex set $V(D)$ is defined as a function$fcolon V(D)to {0, 1, 2}$ such that every vertex $vin V(D)$ with $f(v)=0$ has at least two in-neighborsassigned 1 under $f$ or one in-neighbor $w$ with $f(w)=2$. A set ${f_1,f_2,ldots,f_d}$ of distinctItalian dominating functions on $D$ with the property that $sum_{i=1}^d f_i(v)le 2$ for each $vi...

This paper surveys some of the work that was inspired by Wagner’s general technique to prove completeness in the levels of the boolean hierarchy over NP and some related results. In particular, we show that it is DP-complete to decide whether or not a given graph can be colored with exactly four colors, where DP is the second level of the boolean hierarchy. This result solves a question raised ...

A subset, D, of the vertex set of a graph G is called a dominating set of G if each vertex of G is either in D or adjacent to some vertex in D. The maximum cardinality of a partition of the vertex set of G into dominating sets is the domatic number of G, denoted d(G). G is said to be domatically critical if the removal of any edge of G decreases the domatic number, and G is domatically full if ...

We resolve the problem posed as the main open question in [4]: letting δ(G), ∆(G) and D(G) respectively denote the minimum degree, maximum degree, and domatic number (defined below) of an undirected graph G = (V,E), we show that D(G) ≥ (1−o(1))δ(G)/ ln(∆(G)), where the “o(1)” term goes to zero as ∆(G) → ∞. A dominating set of G is any set S ⊆ V such that for all v ∈ V , either v ∈ S or some nei...