MAT 375 HOMEWORK 1
For each of the following statements, decide if it is true or false, and either
prove it or give a counterexample.
1. Every simple graph with 2 vertices has two vertices with the same
degree.
2. There is a simple graph with 8 vertices t
CSOR E4010: LECTURE NOTES 3
5. Shortest Paths
Let s, t be vertices of a connected graph G, and for each edge e let
w(e) 0 be its length. We wish to nd the path P of G from s to t
with shortest length, that is, with eE(P ) w(e) minimum. We write w(P )
for
CSOR MAT375: LECTURE NOTES 5
11. Matchings and Tuttes Theorem
A matching M covers a vertex v if v is incident with an edge in M . A
matching is perfect (or complete, or a 1-factor) if it covers every vertex of
the graph.
If G is a graph and X V (G), odd (
MAT375 LECTURE NOTES 2
3. Trees, Forests
Forest: Graph with no cycles.
Tree: Non-null connected forest.
(3.1) If G is a forest then comp(G) = |V (G)| |E(G)|.
Proof: Induction on |E(G)|, using (2.7).
The result of deleting e, when e is an edge, is the subg
MAT 375 : LECTURE NOTES 6
13. Planar Graphs
A drawing of a graph G in the plane (or other surfaces, later) means what
you would expect: each vertex of G is represented by a (distinct) point of
the plane, and each edge by a line segment joining the points
CSOR E4010: LECTURE NOTES 1
1. Basic Denitions
Graph G: Consists of a set of vertices (denoted by V (G), a set of edges
(denoted by E(G), and an incidence relation; each edge is incident with
either one or two vertices (its ends).
Adjacent: Distinct verti
MAT 375 HOMEWORK 9
1. Let G be a 2-connected loopless drawn in the plane graph. For each
1
vertex v, dene S(v) = 1 deg(v) . Show that for some region r, vr S(v) <
2
1, where the sum is over all vertices v incident with r. [Hint: what is
the average of vr
MAT 375 HOMEWORK 8
1. What is the maximum possible number of edges in a simple planar
bipartite graph with n 3 vertices? Deduce that K3,3 is not planar.
2. Let G be simple and planar, with no vertex of degree 4. Is it true
that G must have two vertices of
MAT375 HOMEWORK 7
1. Show that every 3-edge-connected cubic graph has a perfect matching.
(Hint: if X V (G), |(X)| 3|X|.) What about 2-edge-connected
cubic graphs?
2. Let G be a graph and Z V (G). Show that the following are equivalent:
(a) G has a matchi
MAT375 HOMEWORK 3
1. X E(G) is even-degree if every vertex of G is incident with an even
number of edges in X. Show that if X and Y are both even-degree
then so is (X Y ) (Y X). Deduce that if T is a spanning tree of
G, there is an even-degree set X E(G)
MAT 375 HOMEWORK 2
1. Let G be a graph with |E(G)| = |V (G)| 1. Show that G is connected
if and only if G has no cycles.
2. Let T be a tree, and let T1 , T2 be connected subgraphs of T with
V (T1 T2 ) = . Show that T1 T2 and T1 T2 are trees.
3. Let T be a
MAT375 HOMEWORK 6
1. Let G be a digraph and for each edge e let (e) 0 be an integer, so
that for every vertex v,
(e) =
e (v)
(e)
e + (v)
Show there is a list C1 , ., Cn of directed cycles (possibly with repetition) so that for every edge e of G,
|cfw_i :
MAT375 HOMEWORK 4
1. Let G be a bipartite graph, with bipartition (A, B). Show that the
following are equivalent:
(a) there is a matching M in G so that every vertex in A is an end
of some member of M
(b) for every X A there are at least |X| vertices in B
MAT375 HOMEWORK 5
Let G be a graph and X V (G). We denote by (X) the set of all
edges of G with one end in X and the other in V (G) X. If D E(G), D
is called a cut if D = (X) for some X V (G) with 0 < |X| < |V (G)|.
1. Show that if C is a cycle of G and D
MAT375: LECTURE NOTES 4
8. Matchings in bipartite graphs
A matching in G is a subset M E(G) so that no edge in M is a loop,
and no two edges in M are incident with the same vertex. If M is a matching
in G, a path P of G is M -alternating if the edges of P