Question

(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.

Answer #2

answered by: anonymous

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.

Exercise 10.5.4: Edge connectivity between two vertices.
Two vertices v and w in a graph G are said to be
2-edge-connected if the removal of any edge in the graph leaves v
and w in the same connected component.
(a) Prove that G is 2-edge-connected if every pair of vertices
in G are 2-edge-connected.

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.

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.

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?

a) Let k>1 be the size of a minimum edge cut in G. Show that
the deletion of k edges from G results in at most 2 components.
b) Is the same true for vertex cuts? Justify your answer.

Let G be a simple graph having at least one edge, and let L(G)
be its line graph. (a) Show that χ0(G) = χ(L(G)). (b) Assume that
the highest vertex degree in G is 3. Using the above, show Vizing’s
Theorem for G. You may use any theorem from class involving the
chromatic number, but no theorem involving the chromatic index

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 e be the unique lightest edge in a graph G. Let T be a
spanning tree of G such that e ∉ T . Prove using elementary
properties of spanning trees (i.e. not the cut property) that T is
not a minimum spanning tree of G.

GRAPH THEORY:
Let G be a graph which can be decomposed into Hamilton
cycles.
Prove that G must be k-regular, and that k must be even.
Prove that if G has an even number of vertices, then the edge
chromatic number of G is Δ(G)=k.

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