Infinite graphs isomorphic to their line graph

Question

The only finite connected graphs $G$ that are isomorphic to their line graph $L(G)$ are the cycle graphs $C_n$ (see this link for example).

There are connected countable graphs that are isomorphic to their line graph:

• $G=(\omega,E)$ where $E=\{\{k,k+1\}: k\in\omega\}$;
• $G=(\mathbb{Z},E)$ where $E=\{\{k,k+1\}: k\in\mathbb{Z}\}$.

Note that in the second graph, all vertices have degree 2, which makes it into a kind of "infinite cycle". (An interesting side question would be whether these are the only connected countable graphs (up to isomorphism) that are isomorphic to their line graph.)

Question. Is there a connected graph $G=(V,E)$ with $V$ uncountable such that $G\cong L(G)$?

Answer 1

Note first that $L$ is naturally an endofunctor on the category of graphs and injective graph-homomorphisms that commutes with filtered colimits. Let $G$ be any graph such that there is an embedding $i:G\to L(G)$. This gives rise to an embedding $L(i):L(G)\to L(L(G))$, an embedding $L(L(i)):L(L(G))\to L(L(L(G)))$, and so on. Let $L^\omega(G)$ be the colimit of $G\to L(G)\to L(L(G))\to \dots$. Since $L$ commutes with filtered colimits, there are canonical isomorphisms $$L(L^\omega(G))\cong L(\varinjlim(G\to L(G)\to\dots))\cong \varinjlim( L(G)\to L(L(G))\to\dots)\cong L^\omega(G).$$

Furthermore, if $G$ is connected, so is $L^\omega(G)$, and if $G$ is infinite, $L^\omega(G)$ has the same cardinality as $G$. Thus to get a connected graph of a given infinite cardinality isomorphic to its line graph, it suffices to give a connected graph $G$ of that cardinality that embeds in $L(G)$. But this is easy; for instance, complete graphs work. In fact, any connected graph whose cardinality $\kappa$ has uncountable cofinality works, since such a graph must contain a vertex of degree $\kappa$, and thus $L(G)$ contains a clique of size $\kappa$.