A Catlin-type theorem for graph partitioning avoiding prescribed subgraphs

As an extension of the Brooks theorem, Catlin in 1979 showed that if H is neither odd cycle nor a complete graph with maximum degree Δ(H), then has vertex Δ(H)-coloring such one color classes independent set. Let G be connected order at least 2. A G-free k-coloring partition set into V1,…,Vk H[Vi], subgraph induced on Vi, does not contain any isomorphic to G. generalization Catlin's Theorem we show ⌈Δ(H)δ(G)⌉-coloring for which subset V(H) satisfies following conditions; (1) regular, (2) Kkδ(G)+1 G≃Kδ(G)+1, and (3) K2. Indeed, even more, by proving G1,…,Gk are graphs minimum degrees d1,…,dk, respectively, Δ(H)=∑i=1kdk, there vertices each H[Vi] Gi-free moreover Vi's can chosen way except either k=1 G1, Gi Kdi+1 KΔ(H)+1, or K2 cycle.