Let V1- is the union of connected dominating sets in all connected components of graph G=(V,E); V2- is the union of connected dominating sets of G-V1; V3- is the union of connected dominating sets of G-V1-V2 and so on. We denote maximum length of such sequence V1, V2, V3,… for all induced subgraphs of G by h(G).
It is clear that h( G ) ≤number of Hadwiger(because the removal of a connected dominating set of vertices reduces the number of Hadwiger). If k=k(G)- chromatic number and k≤4, then h (G)≥k (proved).
Conjecture: h(G)>=k(G) for all graphs G.