Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Does any proof exist that a simple graph with $n$ vertices such that the least vertex degree is $\geq \frac{n-1}{2}$ is a connected graph? (i.e. does such a proof have a name?)

share|cite|improve this question
up vote 2 down vote accepted

Suppose there is a subset of $m$ vertices that form a connected component. What is the max number of edges in this subgraph? (think of it as a complete graph on $m$ vertices)

What is the minimum number of edges in this subgraph? (think of it as an $((n-1)/2)$-regular graph with $m$ vertices)

Compare those two numbers.

Explicitly, the first number is $m(m-1)/2$, and the second is $m(n-1)/2$. You must have $m(n-1)/2 \leq m(m-1)/2$ (min $\leq$ max), and $m \leq n$ (it's a subgraph), so $m=n$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.