Given an Eulerian graph G and 2 edges adjacent on the same vertex. Is there an Eulerian circuit that these two edges are consecutive? ie e1 - V1 - e2.
2026-03-26 22:19:24.1774563564
Consecutive edges on Eulerian Circuit
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(The previous version of this answer was incorrect as it did not consider all cases. Thanks to @D.L for pointing this out in comments)
Equivalent Question: $\forall$ eulerian graph $G$ and given any two edges $\{e_1,e_2\}$ adjacent to a vertex V, can we construct an eulerian path $P$ going through $\{e_1,e_2\}$ consecutively?
Answer: No, we cannot construct such a path. Please see here: If $G$ is an Eulerian graph with edges $a$, $b$ sharing a vertex $v$, is it true that $G$ has an Eulerian trail in which $a$, $b$ are consecutive?