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Is it possible to have a graph with vertices of degrees: 1, 1, 2, 3, 3? How do we tackle such questions?

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    "How do we tackle such questions?" Trial and error, mostly. As long as the sequence obeys the handshaking lemma, there's not much else to do, I think. – 2017-02-20
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    This might help : https://en.wikipedia.org/wiki/Degree_(graph_theory)#Degree_sequence – 2017-02-20
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    @maverick This should answer your question : http://mathworld.wolfram.com/GraphicSequence.html – 2017-02-20

3 Answers 3

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Note that if you allow loops and parallel edges then the problem is very simple: just draw 5 nodes and $1+1+2+3+3=10$ edge ends, then join the edge ends whichever way you like. This is possible as $10$ is even (if the total were odd it would be impossible). enter image description here

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Mostly try and error. What I learned so far is that there are no good characterizations of degree sequences.

enter image description here

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We're not completely empty handed.

Sanity checks:

  • If a sequence has a number greater than $n-1$, where $n$ is the number of coordinates (vertices), then it cannot be constructed (as there's not enough vertices to form the neighbors).

  • If a sequence sums to an odd number, it would violate the Handshaking Lemma, and can't be constructed.

  • If there's a lot of vertices of degree $1$, and two or more vertices of high degree, sometimes this can make it impossible (if we were to attempt to create one, we'd end up with too many vertices of degree $2$ or more).

Proving existence:

Construct example:

Construct all non-isomorphic examples:

  • I tend to just use geng (which comes with Nauty), with the command line geng n e:e where $n$ is the number of vertices (in this case 5) and $e$ is the number of edges (in this case $\tfrac{1}{2}(1+1+2+3+3)=5$). Afterwards, I filter out the ones which don't have the degree sequence using a GAP script.

In this example, there is a unique graph (with vertices marked with their degree):

enter image description here