By Litherland R.
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Since also degG v ≤ k ≤ n − 1 − k, there are at least n − k vertices of degree at most n − k − 1 in G, and dn−k ≤ n − k − 1. 8. A set U of vertices of a graph G is independent if any two distinct elements of U are independent. The independence number of G, β(G), is the maximum number of vertices in an independent set. For example, β(G) = 1 iff G is complete. 9. Let G be a graph of order n ≥ 3. If κ(G) ≥ β(G) then G is Hamiltonian. Proof. Let k = κ(G). Then k ≥ 2, since otherwise β(G) = 1 and G ∼ = Kn , contradicting k = 1.
7. 6, α = (v1 v4 v7 v8 )(v2 v3 )(v5 v9 v10 v6 ) is an automorphism of the Petersen graph with F (α) = (1 2 3 4) ∈ S5 . Vertices u and v of a graph G are similar if there is an automorphism α of G with α(u) = v. This is an equivalence relation on V (G), and the equivalence classes are the orbits of G. A graph with a single orbit is vertex transitive. 2). At the other extreme, a graph is asymmetric if its only automorphism is the identity; equivalently, its orbits are all singletons. 8. There is an asymmetric graph of order n > 1 iff n ≥ 6.
1) holds for some i with 1 ≤ i ≤ n − 2. Ti−1 has one more vertex than Ti , namely vi , which is an end-vertex of Ti−1 and is not in the set {si , . . , sn−2 }. Further, a vertex of Ti is an end-vertex of Ti−1 iff it is an end vertex of Ti and not equal to si , which is the case iff it is not in {si , . . , sn−2 }. 1) for i − 1. Now let (s1 , . . , sn−2 ) be any sequence of elements of X. We construct graphs Gi and sets Xi ⊆ X for 0 ≤ i ≤ n − 2 such that V (Gi ) = X, |E(Gi )| = i, |Xi | = n − i, {si+1 , .
Introduction to graph theory, Lecture notes by Litherland R.
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