Borodin-Kostochka conjecture and Partitioning a graph into classes with no clique of specified size

Abstract

For a given graph $H$ and the graphical properties $P_1, P_2,\ldots,P_k$, a graph $H$ is said to be $(V_1, V_2,\ldots,V_k)$-partitionable if there exists a partition of $V(H)$ into $k$-sets $V_1, V_2\ldots,V_k$, such that for each $i\in[k]$, the subgraph induced by $V_i$ has the property $P_i$. In $1979$, Bollobás and Manvel showed that for a graph $H$ with maximum degree $Δ(H)\geq 3$ and clique number $ω(H)\leq Δ(H)$, if $Δ(H)= p+q$, then there exists a $(V_1,V_2)$-partition of $V(H)$, such that $Δ(H[V_1])\leq p$, $Δ(H[V_2])\leq q$, $H[V_1]$ is $(p-1)$-degenerate, and $H[V_2]$ is $(q-1)$-degenerate. Assume that $p_1\geq p_2\geq\cdots\geq p_k\geq 2$ are $k$ positive integers and $\sum_{i=1}^k p_i=Δ(H)-1+k$. Assume that for each $i\in[k]$ the properties $P_i$ means that $ω(H[V_i])\leq p_i-1$. Is $H$ a $(V_1,\ldots,V_k)$-partitionable graph? In 1977, Borodin and Kostochka conjectured that any graph $H$ with maximum degree $Δ(H)\geq 9$ and without $K_{Δ(H)}$ as a subgraph, has chromatic number at most $Δ(H)-1$. Reed proved that the conjecture holds whenever $ Δ(G) \geq 10^{14} $. When $p_1=2$ and $Δ(H)\geq 9$, the above question is the Borodin and Kostochka conjecture. Therefore, when all $p_i$s are equal to $2$ and $Δ(H)\leq 8$, the answer to the above question is negative. Let $H$ is a graph with maximum degree $Δ$, and clique number $ω(H)$, where $ω(H)\leq Δ-1$. In this article, we intend to study this question when $k\geq 2$ and $Δ\geq 13$. In particular as an analogue of the Borodin-Kostochka conjecture, for the case that $Δ\geq 13$ and $p_i\geq 2$ we prove that the above question is true.