Generalized circular colouring of graphs

Let P be a graph property and r,s ∈ N, r ≥ s. A strong circular (P,r,s)-colouring of a graph G is an assignment f:V(G) → {0,1,...,r-1}, such that the edges uv ∈ E(G) satisfying |f(u)-f(v)| < s or |f(u)-f(v)| > r - s, induce a subgraph of G with the propery P. In this paper we present some basic results on strong circular (P,r,s)-colourings. We introduce the strong circular P-chromatic number of a graph and we determine the strong circular P-chromatic number of complete graphs for additive and hereditary graph properties.