Generalized Petersen graphs are certain graphs consisting of one quadratic factor. For these graphs some numerical invariants concerning the domination are studied, namely the domatic number $d(G)$, the total domatic number $d_t(G)$ and the $k$-ply domatic number $d^k(G)$ for $k=2$ and $k=3$. Some exact values and some inequalities are stated.
Let G1 and G2 be copies of a graph G, and let f : V (G1) → V (G2) be a function. Then a functigraph C(G, f) = (V, E) is a generalization of a permutation graph, where V = V (G1) ∪ V (G2) and E = E(G1) ∪ E(G2) ∪ {uv : u ∈ V (G1), v ∈ V (G2), v = f(u)}. In this paper, we study colorability and planarity of functigraphs.
ct. Guy and Harary (1967) have shown that, for k > 3, the graph P[2k, k] is homeomorphic to the Möbius ladder M2k, so that its crossing number is one; it is well known that P[2k, 2] is planar. Exoo, Harary and Kabell (1981) have shown hat the crossing number of P[2k + 1, 2] is three, for k ≥ 2. Fiorini (1986) and Richter and Salazar (2002) have shown that P[9, 3] has crossing number two and that P[3k, 3] has crossing number k, provided k ≥ 4. We extend this result by showing that P[3k, k] also has crossing number k for all k ≥ 4.