γ(G) denotes the chromatic number of G. G is γ-critical (o: vertex-critical) if the deletion of any vertex of G gives a (γG)-1)-chromatic graph. For A <V(G)c(A)A denotes the number of components after the deletion of the vertices of A. GA is the subgraph of G induced by A. Sq(GA) denotes the number of colorings of GA with at most q colors. We will prove the following results: If G is γ-critical with γ(G)=1+I, then c(A)<-sa(G)A I for every A < V(G). If Go is not a complete graph and q is large enough then there exists a γ-critical G with γ(G)= q + 1 which has an articulation set A satisfying GA ≈ G0 and c(A) = Sq(GA).
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics