Degree sequence of a graph has repeated entries

Question

I am trying to brush up my graph theory skills. I have not done any in over 4 years and i am rusty...If someone could help me out with this simple proof i would appreciate it.

Prove that for any graph $G$ of order at least 2, the degree sequence has at least one pair of repeated entries.

So the degree sequence if a list of the degrees of each vertex (usually written in descending order).

I know that the sum of the degrees of the vertices of a graph is equal to $2|E|$ and that the number of vertices of odd degree is even.

If someone could help me out and point me in the right direction I would appreciate it.

The problem you are referring to is known as the handshaking lemma. — Isaac Kleinman, Mar 06 '12 at 14:46
See If $n$ is a natural number $\ge 2$ how do I prove that any graph with $n$ vertices has at least two vertices of the same degree? and Why do graph degree sequences always have at least one number repeated? — Martin Sleziak, Jun 19 '15 at 09:06

score 7 · Accepted Answer · edited Jun 19 '15 at 09:03

7

Yes, it just came to me after I posted the comment to the previous answer.

A graph with at least 2 vertices and no edges the 0 degree is repeated. A graph with 2 vertices and 1 edge the degree 1 is repeated. Each vertex has at least one entry in the degree sequence, so there are a total of $N$ entries. But each degree can have a maximum of degree = $N-1$. Therefore according to the pigeonhole principle at least one of the degrees has to repeat.

edited Jun 19 '15 at 09:03

Martin Sleziak

53,687

answered Aug 11 '10 at 01:43

gprime

593

And why can't the sequence 0,1,2,...,N-1 be a degree sequence? [signature removed by moderator] – G. Paseman Aug 11 '10 at 17:12
If there is a vertex of degree N-1 in your graph on N vertices, every other vertex has degree at least 1. – Serge Gaspers Sep 01 '10 at 22:18

score 3 · Answer 2 · edited Aug 11 '10 at 22:04

3

If there were no repeats, what would the degree sequence look like? Why could it not look like that?

edited Aug 11 '10 at 22:04

Isaac

36,557

answered Aug 10 '10 at 23:50

G. Paseman

340

1

I'd probably add such hints/questions as a comment rather than an answer--then if the OP figures it out from those he can post a full answer himself. – Jamie Banks Aug 11 '10 at 00:26
so the degree sequence would be unique, obviously, and it cannot look like that because...my mind is drawing a blank... – gprime Aug 11 '10 at 01:35
Perhaps. For this problem, it felt more like an answer. [signature removed by moderator] – G. Paseman Aug 11 '10 at 03:35
@Katie, @G. Paseman: Hints are acceptable as answers when they are hints because the answerer doesn't want to give away the problem – Casebash Aug 11 '10 at 07:55

Degree sequence of a graph has repeated entries

2 Answers2