Let $G_1 = (V, E_1)$ and $G_2 = (V, E_2)$ be two such graphs on the same vertex set $V$ that both have Eulerian circuit. Let $G_3 = (V, E_3)$ be the graph also on vertex set $V$ given by $E_3 = (E_1\setminus E_2) \cup (E_2 \setminus E_1)$, that is two vertices are connected by an edge in $G_3$ if and only if they are connected in either $G_1$ or in $G_2$ but not in both. Is it true that if $G_3$ is connected, then it has an Eulerian circuit?
2026-03-26 18:30:16.1774549816
eulerian circuit combinatorics
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Hint: a connected graph has an Eulerian circuit if and only if all vertices have even degree. So can you show that if $v$ has even degree in $G_1$ and $G_2$ then it also has even degree in $G_3$?