Question

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/∆.

Answer #1

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 or disapprove each of the following:
(a) Every disconnected graph has an isolated vertex.
(b) A graph is connected if and only if some vertex is connected
to all other vertices.
(c) If G is a simple, connected, Eulerian graph, with edges e, f
that are incident to a common vertex, then G has an Eulerian
circuit in which e and f appear consequently.

You are given a directed acyclic graph G(V,E), where each vertex
v that has in-degree 0 has a value value(v) associated with it. For
every other vertex u in V, define Pred(u) to be the set of vertices
that have incoming edges to u. We now define value(u) = ?v∈P red(u)
value(v). Design an O(n + m) time algorithm to compute value(u) for
all vertices u where n denotes the number of vertices and m denotes
the number of edges...

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.

Let G be a simple graph in which all vertices have degree four.
Prove that it is possible to color the edges of G orange or blue so
that each vertex is adjacent to two orange edges and two blue
edges.
Hint: The graph G has a closed Eulerian walk. Walk along it and
color the edges alternately orange and blue.

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

Graph G is a connected planar graph with 1 face. If G is finite,
prove that there is a vertex with degree 1.

Suppose that we generate a random graph G = (V, E) on the vertex
set V = {1, 2, . . . , n} in the following way.
For each pair of vertices i, j ∈ V with i < j, we flip a fair
coin, and we include the edge i−j in E if and only if the coin
comes up heads.
How many edges should we expect G to contain?
How many cycles of length 3 should we...

Graph Theory
Let v be a vertex of a non trivial graph G. prove that if G is
connected, then v has a neighbor in every component of G-v.

Graph Theory.
A simple graph G with 7 vertices and 10 edges has the
following properties: G has six vertices of degree
a and one vertex of degree b. Find a and
b, and draw the graph.
Show all work.

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