Question

Supposed G is a graph, possibly not connected and u is a vertex of odd degree. Show that there is a path from u to another vertex v does not equal u which also has odd degree.(hint: since u has odd degree it has paths to some other vertices. Just consider those.)

Answer #1

Supposed G is a graph, possibly not connected and u is a vertex
of odd degree. Show that there is a path from u to another vertex v
6= u which also has odd degree.(hint: since u has odd degree it has
paths to some other vertices. Just consider those.)

Prove that if G is a connected graph with exactly 4 vertices of
odd degree, there exist two trails in G such that each edge is in
exactly one trail. Find a graph with 4 vertices of odd degree
that’s not connected for which this isn’t true.

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 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 a graph where every vertex has odd degree, and G has a
perfect matching. Prove that if M is a perfect matching of G, then
every bridge of G is in M.
The Proof for this question already on Chegg is wrong

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.

6. If
a graph G has n vertices, all of which but one have odd degree, how
many
vertices of odd degree are there in G, the complement of G?
7.
Showthatacompletegraphwithmedgeshas(1+8m)/2vertices.

Problem 2. Consider a graph G =
(V,E) where |V|=n.
2(a) What is the total number of possible paths of length
k ≥ 0 in G from a given starting vertex
s and ending vertex t? Hint: a path of length
k is a sequence of k + 1 vertices without
duplicates.
2(b) What is the total number of possible paths of any length in
G from a given starting vertex s and ending
vertex t?
2(c) What is the...

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.

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.

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