Vertex Partitions and Maximum $\G$-free Subgraphs

We define a $(V_1, V_2, \ldots, V_k)$-partition for a given graph $H$ and graphical properties $P_1, P_2, \ldots, P_k$ as a partition where each $V_i$ induces a subgraph of $H$ with property $P_i$. Matamala (2007) extended this result by showing that for any graph $H$ with $\Delta(H)=p+q$, there exists a $(V_1, V_2)$-partition of $V(H)$ where $H[V_1]$ is a maximum order $(p-1)$-degenerate induced subgraph and $H[V_2]$ is $(q-1)$-degenerate. Additionally, Catlin and Lai proved that if $\Delta(H)\geq 5$, $H$ has a $(V_1, V_2)$-partition such that $H[V_1]$ is a maximum order acyclic induced subgraph, $\omega(H[V_2])\leq \Delta(H)-2$, and $\Delta(H[V_2])\leq \Delta(H)-2$. Rowshan and Taherkhani demonstrated that given a graph $G$ with a minimum degree $\delta(G)$ and for $k=\lceil \frac{\Delta(H)}{\delta(G)}\rceil$, there exists a $(V_1, V_2, \ldots, V_k)$-partition of the vertex set of $H$, such that each $H[V_i]$ is $G$-free, meaning it does not contain a subgraph isomorphic to $G$, and $H[V_1]$ is a maximum order $G$-free induced subgraph. In our paper, we present a novel result for a connected graph $H$ with $\Delta(H)\geq 5$ and without $K_{\Delta(H)+1}\setminus e$ as a subgraph. We establish that when $p_1\geq p_2\geq\cdots\geq p_{k-1}\geq 2$, $p_k\geq 4$, $\sum_{i=1}^k p_i=\Delta(H)-1+k$, and $\mathcal{G}_i$ represents a family of graphs with a minimum degree at least $p_i-1$ for each $i\in [k-1]$, a $(V_1, V_2, \ldots, V_k)$-partition of $V(H)$ exists. This partition guarantees that $H[V_1]$ is a maximum order $\mathcal{G}_1$-free induced subgraph, $H[V_i]$ is $\mathcal{G}_i$-free for each $2\leq i\leq k-1$, $\Delta(H[V_k])\leq p_k$, and either $H[V_k]$ is $K_{p_k}$-free or its $p_k$-cliques are disjoint.