Let $G$ be graph, let $x\in V(G)$ with $|\delta_G(x)|=\Delta(G)$. For all other nodes $v\in V(G)\setminus\{x\}$ let $|\delta_G(x)|\lt\Delta(G)$. Furthermore assume we have $v_1,v_2,v_3\in V(G)$ such that they are neighbors of $x$ and $\{v_1,v_2,v_3\}$ is a stable set in $G$. Also let $G[V(G)\setminus\{v_1,v_2,v_3\}]$ be connected. Show that we can color the nodes of $G$ using $\Delta(G)-1$ colors.
Since we do not have a complete graph nor an odd cycle, we know that $\chi(G)\le\Delta(G)$. Now if we look at $G':=G[V(G)\setminus\{x\}]$, we see that we can color $G'$ using $\Delta(G)-1$ colors. Now I would be done if any pair of $v_1,v_2,v_3$ had the same color, but I am not able to come up with a convincing argument.