Show that $G$ is a simple graph and $m> {n-1 \choose 2}$, then $G$ is connected.

Question

I'm having trouble with this problem. I'm using Bondy and Murty's Graph Theory book and i cannot continue to prove it. Here is what i've started:

Let $G$ be a graph. From the appendix of Bondy and Murty's Graph Theory which says if $G$ is not connected, we can partition the vertices into parts $(X,Y)$ such that no edge joins a vertex in $X$ to a vertex in $Y$. Hence, we have at least $|X|$ $\cdot$ $|Y|$. I'm stuck here and i don't know the next step, please help me prove this.

I'm confused where to start since I'm also new to this theory. Can I ask sone help from you? Thank you for your time of considering my problem. God bless.

As a starting point, do you know what the maximum number of edges a simple graph with $n$ vertices can have? — Brian Moehring, Jul 30 '21 at 06:30

score 1 · Answer 1 · answered Jul 30 '21 at 11:56

1

If $G$ is disconnected, then we can divide the set $V$ of vertices into two subsets $V_1$ and $V_2$ such that there is no edges between $V_1$ and $V_2$. Hence, we have at least $|V_1|\cdot |V_2|$ "non-edges". Can you continue now?

answered Jul 30 '21 at 11:56

richrow

4,727

Show that $G$ is a simple graph and $m> {n-1 \choose 2}$, then $G$ is connected.

1 Answers1