Question

Let G be a connected graph. Show that G has a subtree that is a maximal tree.

Answer #1

Let G be a connected plane graph and let T be a spanning tree of
G. Show that those edges in G∗ that do not correspond to the edges
of T form a spanning tree of G∗ . Hint: Use all you know about
cycles and cutsets!

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!

Show that an edge e of a connected graph G belongs to any
spanning tree of G if and only if e is a bridge of G. Show that e
does not belong to any spanning tree if and only if e is a loop of
G.

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.

Let G = (X, E) be a connected graph. The distance between two
vertices x and y of G is the shortest length of the paths linking x
and y. This distance is denoted by d(x, y). We call the center of
the graph any vertex x such that the quantity max y∈X d(x, y) is
the smallest possible. Show that if G is a tree then G has either
one center or two centers which are then neighbors

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.)

(a) Let L be a minimum edge-cut in a connected graph G with at
least two vertices. Prove that G − L has exactly two
components.
(b) Let G an eulerian graph. Prove that λ(G) is even.

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.

Suppose G is a simple, nonconnected graph with n vertices that
is maximal with respect to these properties. That is, if you tried
to make a larger graph in which G is a subgraph, this larger graph
will lose at least one of the properties (a) simple, (b)
nonconnected, or (c) has n vertices.
What does being maximal with respect to these properties imply
about G?G? That is, what further properties must GG possess because
of this assumption?
In this...

Let G be a graph on 10 vertices that has no triangles. Show that
Gc (G complement) must have K4 subgraph

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