Partitioning of a graph into induced subgraphs not containing prescribed cliques

Abstract

Let Kp be a complete graph of order p≥2. A Kp-free k-coloring of a graph H is a partition of V(H) into V1,V2…,Vk such that H[Vi] does not contain Kp for each i≤k. In 1977 Borodin and Kostochka conjectured that any graph H with maximum degree Δ(H)≥9 and without KΔ(H) as a subgraph has chromatic number at most Δ(H)−1. As analogue of the Borodin–Kostochka conjecture, we prove that if p1≥⋯≥pk≥2, p1+p2≥7, ∑i=1kpi=Δ(H)−1+k, and H does not contain KΔ(H) as a subgraph, then there is a partition of V(H) into V1,…,Vk such that for each i, H[Vi] does not contain Kpi. In particular, if p≥4 and H does not contain KΔ(H) as a subgraph, then H admits a Kp-free ⌈Δ(H)−1p−1⌉-coloring. Catlin showed that every connected non-complete graph H with Δ(H)≥3 has a Δ(H)-coloring such that one of the color classes is maximum K2-free subset (maximum independent set). In this regard, we show that there is a partition of vertices of H into V1 and V2 such that V1 is a maximum Kp-free subset of H and V2 is a Kq-free subset of H, if p≥4, q≥3, p+q=Δ(H)+1, and its clique number ω(H)=p.