A Catlin-type theorem for graph partitioning avoiding prescribed subgraphs

Abstract

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