MT4514: Graph Theory

12 Colva M. Roney-DougalGiven a sequence of non-negative integers, how can we tell whether there is agraph with that as a degree sequence? For example, 6, 6, 5, 5, 3, 3, 3, 3 is the degreesequence of a simple graph, but 6, 6, 5, 5, 2, 2, 2, 2 is not.Theorem 4.2. The following are necessary conditions for d 1 , . . . , d v to be a degreesequence.1. The sum of the d i is even.2. We have d i ≤ v − 1 for 1 ≤ i ≤ v.3. At least two vertices have equal degree.Proof.1. The sum of the degrees is twice the number of edges (recall the HandshakingLemma, Chapter 1, 1.8).2. In a simple graph, each vertex can be joined to at most v − 1 others.3. Suppose not, then by (2) the degrees must be v − 1, v − 2, . . . , 1, 0. Thereforethe vertex of degree v − 1 must be joined to all other vertices, including thatof degree 0.Remark 4.3. Nonisomorphic graphs can have the same degree sequences.example consider the following two graphs:ForThey both have degree sequence 3, 3, 2, 2, 2 but are nonisomorphic since the firstcontains a triangle but the other does not.Lemma 4.4. Let a graph G have the degree sequences ≥ v 1 ≥ v 2 ≥ · · · ≥ v s ≥ d 1 ≥ · · · ≥ d kand vertices S, V i and D j of corresponding degrees. Then there exists a graph Hwith the same degree sequence such that S is joined to V 1 , V 2 , . . . , V s .Proof. Suppose that in G, the vertex S is not joined to at least one of the V i .Then S must be joined to at least one D j . Recall that v i ≥ d j . If v i = d j then wemay swap the labels V i and D j on the graph to get a graph with S joined to onemore of the V i and one less of the D j vertices.Otherwise v i > d j . Then there is some vertex W joined to V i but not to D j .Consider the graph G ′ obtained from G by removing edges SD j and W V i andreplacing them by edges SV i and W D j :