for a graph to be an Eulerian
1) it must start and end at same vertex with each edge covered exactly once and
2) the degree of each node must be of even degree.
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for a graph with #vertex= 6
possible degree values(to satisfy if its euler or not even degree sequences)=
{2,2,2,2,2,2} , {4,2,2,2,2,2}, {4,4,2,2,2,2} , { 4,4,4,2,2,2} , { 4,4,4,4,2,2} , {4,4,4,4,4,4} = #sequences =7 out of which only 3rd sequence {4,4,2,2,2,2} will have 2 different graphs.
#total =8
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you can use Havel hakimi algorithm to check the validity of the degree sequence... i found each valid.
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i tried each possible sequence but every graph i went for was ISOMORPHIC to the down mentioned 8 graphs.
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