Work out graph degree sequence instantly with clear inputs, formula shown and shareable results.
The handshaking lemma says the degrees sum to twice the number of edges, so an odd total is impossible. Euler's condition then decides traversability: all degrees even gives a closed circuit, exactly two odd gives an open path.
Handshaking lemma
Σ deg(v) = 2·|E|
Eulerian condition
circuit ⇔ every degree even; path ⇔ exactly two odd degrees
The sum is 12, so there are 6 edges, and with two odd degrees an Eulerian path exists but no circuit.
Every edge contributes 1 to each of its two endpoints, so the total is always exactly twice the edge count.