[Math] A closed even walk $W$ need not contain a Cycle.

Question

While going through Graph theory by West ,I learnt that

A closed odd walk $W$ contains an odd cycle.

Proved as

• If $W$ does not contain any repeated vertex ,it is as simple as to state the Theorem.

• In case of a repeated vertex $v$,Break walk into two part both Walk $W_{1}$ and $W_{2}$ starts with $v$ where $W_{1}$,$W_{2}$ $\varepsilon $ $W$
  Now as it is closed odd walk ,we have combination possible

  odd=$\text{even+odd}$

so $W$ must contain an odd Walk having non repeated vertex ,thus it contains odd cycle.

Now My point is that can't we apply the same logic for even closed walk

i.e

even=$\text{even+even}$

even=$\text{odd+odd}$

Then why he is saying that there may not be aycle in closed even walk.

can anyone give counter graph??

can anyone correct me where i am wrong ??

Answer 1

Consider the graph $K_2$. A closed walk is necessarily even in length here. However, there is clearly no cycle here.