Let $G, H$ be two finite, simple, undirected graphs. We call them "reduce-by-1"-isomorphic, or $r_1$-isomorphic for short, if there is a bijection $\psi: V(G) \to V(H)$ such that for all $v \in V(G)$ we have that $G \setminus \{v\}$ is isomorphic to $H \setminus \{\psi(v)\}$.

(Ulam's reconstruction conjecture, or some version of it, states that $r_1$-isomorphic implies isomorphic.)

Can we prove that if $G, H$ are $r_1$-isomorphic then they have the same chromatic number?

EDIT: Will ask about Hadwiger number in different post, it's better to ask one question per post.


The chromatic polynomial, and therefore the chromatic number, was proved reconstructible by Tutte in his famous paper "All the king's horses". I don't know about the Hadwiger number.


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