If $G=(V,E)$ is a simple, undirected graph, then we define its Hadwiger graph $\textrm{Hadw}(G)$ in the following way:

  • $V(\textrm{Hadw}(G)) = \{S\subseteq (V(G): S\neq \emptyset\textrm{ and } S \textrm{ is connected}\}$;
  • $E(\textrm{Hadw}(G)) = \{\{S,T\}\subseteq V(\textrm{Hadw}(G)): S\cap T = \emptyset \textrm{ and } (\exists s\in S, t\in T): \{s,t\}\in E(G) \}.$

$G$ embeds to $\textrm{Hadw}(G)$ by $v\mapsto \{v\}$, and the Hadwiger conjecture states $\chi(G) \leq \omega(\textrm{Hadw}(G))$.

Question: If $G, H$ are infinite connected graphs, does $\textrm{Hadw}(G) \cong \textrm{Hadw}(H)$ imply $G\cong H$?

(This is a follow-up to Isomorphic Hadwiger graphs.)


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.