Connected Domatic Number in Planar Graphs

Connected Domatic Packings in Node-capacitated Graphs

A set of vertices in a graph is a dominating set if every vertex outside the set has a neighbor in the set. A dominating set is connected if the subgraph induced by its vertices is connected. The connected domatic partition problem asks for a partition of the nodes into connected dominating sets. The connected domatic number of a graph is the size of a largest connected domatic partition and it...

The Domatic Number Problem in Interval Graphs

The Domatic Number of Regular Graphs

The domatic number of a graph G is the maximum number of dominating sets into which the vertex set of G can be partitioned. We show that the domatic number of a random r-regular graph is almost surely at most r, and that for 3-regular random graphs, the domatic number is almost surely equal to 3. We also give a lower bound on the domatic number of a graph in terms of order, minimum degree and m...

Chromatic Transversal Domatic Number of Graphs

The chromatic number χ(G) of a graph G is the minimum number of colours required to colour the vertices of G in such a way that no two adjacent vertices of G receive the same colour. A partition of V into χ(G) independent sets (called colour classes) is said to be a χpartition of G. A set S ⊆ V is called a dominating set of G if every vertex in V − S is adjacent to a vertex in S. A dominating s...

The number of polyhedral (3-connected planar) graphs

Data is presented on the number of 3-connected planar graphs, isomorphic to the graphs of convex polyhedra, with up to 26 edges. Results have been checked with the the number of rooted c-nets of R.C. Mullin and P.J. Schellenberg and Liu Yanpei.