# Given a line graph $L(G)$, is it possible to determine whether $G$ is bipartite?

Suppose I am given a line graph $L(G)$, but don't know $G$. Is it possible for me to determine from this information whether $G$ is bipartite?

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Let $G$ be such that $E(G)=\{ \{1,2\},\{2,3\},\{1,3\}\}$ and $V(G)=\{1,2,3\}$. Then the line graph is of the form $V(L(G))=\{a,b,c\}$ and $E(L(G))=\{\{a,b\},\{b,c\},\{a,c\}\}$.
Let $H$ be such that $E(H)=\{\{1,2\},\{1,3\},\{1,4\}\}$ and $V(H)=\{1,2,3,4\}$. Then the line graph is of the form $V(L(H))=\{a,b,c\}$ and $E(L(H))=\{\{a,b\},\{b,c\},\{a,c\}\}$.
$G$ is not bipartite, but $H$ is bipartite, yet they have the same line graph.
This is the only connected counterexample. For example "two copies of $G$" (not bipartite) and "two copies of $H$" (bipartite) have the same line graph too. – Douglas S. Stones Jan 6 '13 at 6:12