Graph Theory(Eular walk)

Question

$A)$ If a graph has closed Eularian walk, then it has an even number of edges

$B)$ If $G$ be a simple graph on $9$ vertices and the sum of all degrees in $G$ is atleast $27$, then $G$ has a vertex of degree atleast $4$.

Which Statement should be true?

Is it possible B) to be true?

And for A) I think "only if" is needed in place of "if" to be true

Comments

''min_degree * no of vertices >= 28 '' Does this mean that we are assuming all the vertices of the graph to have degree equal to the minimum degree?And what does that signify  ? Can someone please explain...

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Here the sum of all the degrees is atleast 28 (it can't be 27 since sum of all degrees have to be even for a graph).

 If we assume that all the graph have minimum degree 3,then sum of the degrees will be (3*9)=27.Hence there has to be a vertex with degree 4.

This is nothing but an application of the extended pigeonhole principle.

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ok,got it,thanks!!!