MA4235 Topics in Graph Theory
Louxin Zhang
Dept of Mathematics
[email protected]
Course Details
Syllabus
- Connectivity
- Domination
- Coloring and planar graphs
- Eulerian digraphs and oriented trees
- (Optional) Ramsey theory
- Electrical networks,
- Pe

Partial Solutions of T5
Q1. It is false. Bipartite graphs can have a very large degree. However, they are always 2colorable.
Q2. Solution 1:
Let F be a k-coloring of G, in which each vertex is assigned an integer in the range from 1 to k.
For each j in cf

Q1. (i) = cfw_(, ), (, ), (, ), (, ), (, ) is a perfect matching.
So, it clearly a maximum matching.
(ii) = cfw_, , , is a maximum independent set. Two reasons for the fact: (1) Any
independent set contains z cannot contain any elements in its neighbor,

MA4235 Topics in Graph Theory
Tutorial 2
1. In a connected graph G, the length of a path is defined to be the number of edges in the path. The
distance between two vertices is the shortest length of a path from one vertex to the other and the
diameter G i

MA4235 Topics in Graph Theory
Tutorial 5
1. Let G be a graph in which each vertex is of degree at least 3. Prove or give a counterexample to the assertion that at least 3 colors are needed to color the vertices of G.
2. Let G be a k-chromatic graph. Show

Selected Solutions to Tutorial 1
2. (a) We first check that the removal of any two vertices u and v does not disconnect
3 . Since 3 is highly symmetric, we can assume u=(0, 0, 0) and only consider the
following cases:
(1) v=(0, 0, 1), (0, 1, 0), or (0, 0

Q1. (a) The following 4 circled vertices form a dominating set.
Hence, () 4. On the other hand, since each vertex has degree 3, any dominating set
must contain at least 4 vertices (See Q4). This concludes that the domination number is 4.
(b) The dominatin

MA4235 Topics in Graph Theory
Tutorial 3
1. Consider the following graph.
Find in G (i) a maximum matching, (ii) a maximum independent set, and (iii) a minimum
vertex cover.
2. List all trees with at most 8 vertices that have a perfect matching.
3. Prove

MA4235 Topics in Graph Theory
Tutorial 4
1. Find the domination number (), independent domination number () and independence
number () of the following graph G.
2. Let G be the following graph and S = cfw_a, c, g. Find all the private neighbours of a, c,

MA4235 Topics in Graph Theory
Tutorial 10
1. Find the eigenvalues of the adjacency matrix and the Laplacian matrix of the complete
bipartite graph 2,3.
2. Compute the eigenvalues of the adjacency matrix of the following graph.
3. Let G be a graph and let

MA4235 Topics in Graph Theory
Tutorial 9
Q1. Let T be a spanning tree of a connected graph G, then, (i) each cut set of G has an edge in
common with T; (ii) each cycle of G has an edge in common with E(G) - E(T).
Q2. How many spanning trees does the label

MA4235: Graph Theory
Louxin Zhang
[email protected]
Domination
Consider a connected graph G=(V, E).
is a dominating set (DS) for G if each
vertex in V-S is adjacent to a vertex in S
f
b
c
a
e
d
g
h
- cfw_a, c, f is not a DS
- cfw_b, f, cfw_a, g, f are DS

MA4235 Topics in Graph Theory
Louxin Zhang
[email protected]
Matching
Consider a (undirected) graph G=(V, E).
Two edges are adjacent if they are
incident to a common vertex.
A matching is a set of pairwise non-adjacent edges.
M is a matching. If , , u

Computing the maximum matching
Berge Theorem: G is a graph. M is not a maximum
matching in G if and only if there is an augmenting path.
How to find an augmenting path of M in
an efficient manner if M is not maximum?
Let u be a vertex not covered by M.
a)

MA4235: Graph Theory
Louxin Zhang
[email protected]
Chapter 4: Edge Coloring
Consider a connected graph G=(V, E), an integer k.
A k-edge-coloring of G is a mapping from E to cfw_1, 2, , k
satisfying: for 1 = , and 2 = , ,
if , , , 1 2 .
i.e., the edges w

MA4235 Topics in Graph Theory
Assignment 2
Submission Deadline: 25 Oct. 2016
Question 1. (a) Let G be a connected graph with at least two vertices. For a subset X of
vertices in G, define () = cfw_ () | (, ) () for some .
Assume = cfw_1 , 2 , , = cfw_1 ,

MA4235 Topics in Graph Theory
Assignment 1
Submission deadline: 20 Sept. 2016
Question 1. Recall that a graph G is bipartite if its vertex set can be split into two
disjoint sets A and B so that each edge joins a vertex of A and a vertex of B.
(a)
(b)
Pro

MA4235 Topics in Graph Theory
Assignment 3
Submission deadline: 8 Nov. 2016
Question 1 A graph G is said to be edge-colouring critical if (i) G is of class 2 and (ii)
( ) < () for any edge e of G. Show that every edge-colouring critical graph does
not co

MA4235 Topics in Graph Theory
Tutorial 7
1. Compute the chromatic index of the Petersen graph (below).
2. A line graph L(G) of a graph is the graph whose vertices are in one-one correspondence with
the edges of G, with two vertices of L(G) being connected

MA4235 Topics in Graph Theory
Tutorial 8
1. A graph G is said to be critical if () < () for any proper subgraph H of G.
Show that 4 constructed on the slide 6 of the lecture on perfect graphs is critical.
2. Show that the octahedron graph (below) is a per

MA4235 Topics in Graph Theory
Tutorial 6
1. Compute the chromatic index of the Petersen graph (below).
2. A line graph L(G) of a graph is the graph whose vertices are in one-one correspondence with
the edges of G, with two vertices of L(G) being connected

MA4235 Topics in Graph Theory
Tutorial 1
1. Let G = (V, E) be a connected graph. For any subset , G[S] denotes the subgraph
induced by S, having the vertex set S and the edge set cfw_(, ) | , .
(a) Is it true in general that () () for each connected induc

MA4235 Topics in Graph Theory
Tutorial 10
1. Compute the current in each edge of the following network when a unit voltage is inserted in
(x, y) and set the resistance in each edge equal to 1.
2. Using Kirchhoffs two laws and the spanning tree with edges

MA4235: Graph Theory
Louxin Zhang
[email protected]
Map graph:
vertices: regions
edges, representing
adjacency
Coloring graphs
= Coloring maps
FACT A Every planar graph does not contain K5 as a subgraph.
Proof. It derived from (1) The following graph K5 i

Vertex Covers in Bipartite Graphs
Consider a (undirected) graph G=(V, E).
A set Q of vertices is called a
vertex cover of G if every edge is
incident with a vertex in Q.
V(G) is a vertex cover of G.
A vertex cover is minimum if it contains as few verti

MA4235: Graph Theory
Louxin Zhang
[email protected]
Matching
Consider a (undirected) graph G=(V, E).
Two edges are adjacent if they are
incident with a common vertex.
A matching is a set of pairwise non-adjacent edges.
M is a matching. If (u, v) is in

Application 2: Counting walks
Let G be a graph with n vertices. , .
- A walk of length k from u to v is a vertex sequence
= 1 , 2 , , +1 =
such that , +1 for j=1, , k.
Theorem A For any k>0, the (i,j)-entry of the
matrix ( ) is equal to the number of wa