Question

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

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.)

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.)

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.

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 ? be a connected graph with at least one edge.
(a) Prove that each vertex of ? is saturated by
some maximum matching in ?.
(b) Prove or disprove the following: Every edge
of ? is in some maximum matching of ?.

Prove that if k is odd and G is a k-regular (k −
1)-edge-connected graph, then G has a perfect matching.

Proof: Let G be a k-connected k-regular graph. Show that, for
any edge e, G has a perfect matching M such that e ε
M.
Please show full detailed proof. Thank you in advance!

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.

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 graph in which there is a cycle C odd length that has
vertices on all of the other odd cycles. Prove that the chromatic
number of G is less than or equal to 5.

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