Question

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).

Answer #1

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/∆.

Let G be a simple graph with n(G) > 2. Prove that G is
2-connected iff for every set of 3 distinct vertices, a,
b and c, there is an a,c-path
that contains b.

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.

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=(V,E) be a connected graph with |V|≥2
Prove that ∀e∈E the graph G∖e=(V,E∖{e}) is disconnected, then G
is a tree.

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.

Let G = (V,E) be a graph with n vertices and e edges. Show that
the following statements are equivalent:
1. G is a tree
2. G is connected and n = e + 1
3. G has no cycles and n = e + 1
4. If u and v are vertices in G, then there exists a unique path
connecting u and v.

A spanning tree of connected graph G = (V, E) is an acyclic
connected subgraph (V, E0 ) with the same vertices as G. Show that
every connected graph G = (V, E) contains a spanning tree. (It is
the connected subgraph (V, E0 ) with the smallest number of
edges.)

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.

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.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 2 minutes ago

asked 21 minutes ago

asked 27 minutes ago

asked 30 minutes ago

asked 33 minutes ago

asked 44 minutes ago

asked 45 minutes ago

asked 53 minutes ago

asked 53 minutes ago

asked 55 minutes ago

asked 55 minutes ago

asked 55 minutes ago