Suppose $G, H$ are finite, simple, undirected graphs and there is a bijection between the vertex sets $\varphi:V(G) \to V(H)$ such that for all $v\in V$ we have $$\text{deg}\_G(v) = \deg\_H(\varphi(v)).$$


Does this imply that $\chi(G) = \chi(H)$?



Do degrees determine the chromatic number?
Let $G$ be a 6-cycle and let $H$ be two 3-cycles.


Or, if you want them connected, let $G$ be a 6-cycle with an extra edge between vertices 1 and 3, and $H$ a 6-cycle with an extra edge between vertices 1 and 4.


