Partitioning of a graph into induced subgraphs not containing prescribed cliques

Abstract

Let $K_p$ be a complete graph of order $p\geq 2$. A $K_p$-free $k$-coloring of a graph $H$ is a partition of $V(H)$ into $V_1, V_2\ldots,V_k$ such that $H[V_i]$ does not contain $K_p$ for each $i\leq k $. In 1977 Borodin and Kostochka conjectured that any graph $H$ with maximum degree $\Delta(H)\geq 9$ and without $K_{\Delta(H)}$ as a subgraph has chromatic number at most $\Delta(H)-1$. As analogue of the Borodin-Kostochka conjecture, we prove that if $p_1\geq \cdots\geq p_k\geq 2$, $p_1+p_2\geq 7$, $\sum_{i=1}^kp_i=\Delta(H)-1+k$, and $H$ does not contain $K_{\Delta(H)}$ as a subgraph, then there is a partition of $V(H)$ into $V_1,\ldots,V_k$ such that for each $i$, $H[V_i]$ does not contain $K_{p_i}$. In particular, if $p\geq 4$ and $H$ does not contain $K_{\Delta(H)}$ as a subgraph, then $H$ admits a $K_p$-free $\lceil{\Delta(H)-1\over p-1}\rceil$-coloring. Catlin showed that every connected non-complete graph $H$ with $\Delta(H)\geq 3$ has a $\Delta(H)$-coloring such that one of the color classes is maximum $K_2$-free subset (maximum independent set). In this regard, we show that there is a partition of vertices of $H$ into $V_1$ and $V_2$ such that $H[V_1]$ does not contain $K_{p}$, $H[V_2]$ does not contain $K_{q}$, and $V_1$ is a maximum $K_p$-free subset of V(H) if $p\geq 4$, $q\geq 3$, $p+q=\Delta(H)+1$, and its clique number $\omega(H)=p$.