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# GATE | GATE-CS-2007 | Question 23

Which of the following graphs has an Eulerian circuit?

(A)

The complement of a cycle on 25 vertices

(B)

A complete graph on 90 vertices

(C)

Any k-regular graph where kis an even number.

(D)

None of the above

Explanation:

A graph has Eulerian Circuit if following conditions are true.

….a) All vertices with non-zero degree are connected. We don’t care about vertices with zero degree because they don’t belong to Eulerian Cycle or Path (we only consider all edges).
….b) All vertices have even degree.

Let us analyze all options.
A) Any k-regular graph where k is an even number. is not Eulerian as a k regular graph may not be connected (property b is true, but a may not)

B) A complete graph on 90 vertices is not Eulerian because all vertices have degree as 89 (property b is false)

C) The complement of a cycle on 25 vertices is Eulerian. In a cycle of 25 vertices, all vertices have degree as 2. In complement graph, all vertices would have degree as 22 and graph would be connected.

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