2 problems

Let $G$ be a graph. For each vertex $v\in V(G)$, let $d(v)$ be its degree and let $\omega(v)$ be the size of the largest clique containing $v$. Write $\chi(G)$ for the chromatic nu…

Let $G$ be a graph and let $L=(L(v):v\in V(G))$ be a list-assignment. For each vertex $v$, let $d(v)$ denote its degree and let $\omega(v)$ denote the size of the largest clique co…