Question

If G is an n-vertex r-regular graph, then show that n/(1+r)≤ α(G) ≤n/2.

Answer #1

Let G be an n-vertex graph with n ≥ 2 and δ(G) ≥ (n-1)/2. Prove
that G is connected and that the diameter of G is at most two.

a
graph is regular of degree k if every vertex has the same degree,
k. show that G has a hamiltonian circuit if G has 13 vertices and
is regular of degree 6.

Let G be a graph or order n with independence number α(G) =
2.
(a) Prove that if G is disconnected, then G contains K⌈ n/2 ⌉ as
a subgraph.
(b) Prove that if G is connected, then G contains a path (u, v,
w) such that uw /∈ E(G) and every vertex in G − {u, v, w} is
adjacent to either u or w (or both).

Prove the following bound for the independence number.
If G is a n-vertex graph with e edges and maximum degree ∆ >
0, then
α(G) ≤ n − e/∆.

a. A graph is called k-regular if every vertex has degree k.
Explain why any k-regular graph with 11 vertices must conain an
Euler circuit. (Hint – think of the possible values of k). b. If G
is a 6-regular graph, what is the chromatic number of G? Must it be
6? Explain

A K-regular graph G is a graph such that deg(v) = K for all
vertices v in G. For example, c_9 is a 2-regular graph, because
every vertex has degree 2. For some K greater than or equal to 2,
neatly draw a simple K-regular graph that has a bridge. If it is
impossible, prove why.

Let G be a graph with vertex set V. Define a
relation R from V to itself as follows: vertex
u has this relation R with vertex v,
u R v, if there is a path in G from u to
v. Prove that this relation is an equivalence relation.
Write your proof with complete sentences line by line in a logical
order. If you can, you may write your answer to this
question directly in the space provided.Your presentation
counts.

Show that if G is connected with n ≥ 2 vertices and n − 1 edges
that G contains a vertex of degree 1.
Hint: use the fact that deg(v1) + ... + deg(vn) = 2e

Show that if G is a graph with n ≥ 2 vertices then G has two
vertices with the same degree.

let G be a connected graph such that the graph formed by
removing vertex x from G is disconnected for all but exactly 2
vertices of G. Prove that G must be a path.

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