Question

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*.

Answer #1

Let G be a connected simple graph with n vertices and m edges.
Prove that G contains at least m−n+ 1 different subgraphs
which are polygons (=circuits). Note: Different polygons
can have edges in common. For instance, a square with a diagonal
edge has three different polygons (the square and two different
triangles) even though every pair of polygons have at least one
edge in common.

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).

Let
G be a simple graph with at least two vertices. Prove that there
are two distinct vertices x, y of G such that deg(x)= deg(y).

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 u and v be distinct vertices in a graph G. Prove that there
is a walk from ? to ? if and only if there is a path from ? to
?.

I.15: If G is a simple graph with at least two vertices, prove
that G has two vertices of the same degree.
Hint: Let G have n vertices. What are possible
different degree values? Different values if G is connected?

Suppose we are going to color the vertices of a connected planar
simple graph such that no two adjacent vertices are with the same
color.
(a) Prove that if G is a connected planar simple graph, then G
has a vertex of degree at most five.
(b) Prove that every connected planar simple graph can be
colored using six or fewer colors.

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 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 δ ≥ 2. Prove that every simple graph G satisfying δmin(G) ≥
δ and
containing no triangles contains a cycle of length at least
2δ.
Prove that this result is sharp by showing that we cannot guarantee
the existence of
a cycle of length at least 2δ + 1. Give a counterexample for each
δ.

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