Undirected graph contains a cycle

Question

How do I prove that the connected undirected graph having 10 nodes and 10 edges contains a cycle.

What do you mean by "the" connected undirected graph having $10$ vertices and $10$ edges? Do you have a specific graph in mind, or are you asking how to prove this for any connected undirected graph having $10$ vertices and $10$ edges? — Ben Sheller, Oct 31 '15 at 17:02

happymath · Accepted Answer · 2015-10-31T16:45:54.653

3

Hint: Any tree with $n$ vertices can have atmost $n-1$ edges.

edited Oct 31 '15 at 16:45

answered Oct 31 '15 at 16:42

happymath

you mean to say $n-1$ edges – TechJ Oct 31 '15 at 16:44
@TechJ yes I hope you got it – happymath Oct 31 '15 at 16:46
Thanks for your reply, I have framed the answer using hint provided by you. – TechJ Oct 31 '15 at 16:51

score 0 · Answer 2 · answered Oct 31 '15 at 16:50

Assume that it is not cyclic, so then it means it is a tree.

But for tree we have number of edges 1 less than number of vertices i.e. $n=n-1$

$n=10-1=9$ which contradicts the given statement.

Hence our assumption is wrong, so there exists a cycle.

2 Answers2