Chapter 4 Vertex Colourings
1. Chromatic Number
2. Greedy Colouring Algorithm (GCA)
Theorem 1. For any graph G, (G) (G) + 1.
Remark. By introducing an appropriate vertex ordering in GCA, the
following result can be proved (will be discussed in tutorial).
Chapter 3 Domination
1. Dominating Sets and Minimal Dominating Sets
In this section, we introduce the concepts of dominating sets and minimal
dominating sets of a graph, and present three fundamental results on these
sets that are due to Ore (1962).
Let G
Chapter 1 Connectivity
1. The vertex- and edge-connectivity
Let G be a connected graph of order n 2 1. A proper subset S of V( G) is called a cut
of G if G S is disconnected. Thus. a vertex w is a cut-vertex of G ifi" cfw_w is a cut. The
vertex-connectivi
Chapter 5
Digraphs
1. Basic Concepts
A directed graph, or simply, a digraph D, is a finite nonempty set V
together with a (possibly empty) set E of ordered pairs of distinct
elements of V. The set V is called the vertex set of D and often denoted
by V(D).
Chapter 2 Matching, Covering and
Independence
1. Matchings in a Graph
Let G be a graph. A subset M of E(G) is called a matching in G if no two
edges in M are adjacent in G.
Lat M be a matching in G. A vertex v in G is said to be M-saturated if v is
incide
MA4235 (Sem 1, 2013/14)
Tutorial 9
1. Let D be a (strong) digraph and v V(D). The eccentricity of v, denoted by
e(v), is defined by
e(v) = maxcfw_d(v, x) : x V(D).
The diameter of D, denoted by d(D), is defined by
d(D) = maxcfw_e(v) : v V(D) = maxcfw_d(x,
MA4235 (Sem 1, 2013/14)
Tutorial 8
1. Determine if the following graphs are planar. Justify your answers.
2. Let G be a connected planar graph, and let n, m and f denote, respectively,
the order, the size and the number of faces in a plane drawing of G. A
MA4235 (Sem 1, 2013/14)
Tutorial 4
1. Consider the following graph G :
Find in G (i) a maximum matching,
(ii) a maximum independent set,
(iii) a minimum v-cover and
(iv) a minimum e-cover.
Hence find the values of (G), (G), (G) and (G).
2. Let G(X, Y) be
MA4235 (Sem 1, 2013/14)
1.
2.
3.
4. Let T be a tournament.
Tutorial 10
5. Let T be a tournament. Show that
(i) any vertex in T which is not a source is dominated by a 2-king;
(ii) if k2(T) 1, then k2(T) 3.
6. Construct
(i) a tournament of order 5 in which
MA4235 (Sem 1, 2013/14)
Tutorial 11
1. (a) Let H be the graph shown below. Find the value of (H).
(b) Let G be a graph obtained from H by adding h new edges to H.
(i) Find the minimum value of h such that (G) = (H) + 1.
(ii) Construct one such G with mini
MA4235 (Sem 1, 2013/14)
Tutorial 7
1. Let G be a non-trivial connected graph. Show that
(i) (G) 1 (G v) (G) for each vertex v in G;
(ii) (G) 1 (G e) (G) for each edge e in G.
2. Draw the graph C5 C4. Find its chromatic number and show a
corresponding colo
MA4235 (Sem 1, 2013/14)
Tutorial 2
1. A graph G contains the following graph as a spanning subgraph.
(i) What is the minimum size m' of G if '(G) 2?
Construct one such graph G of size m'.
(ii) What is the minimum size m* of G if '(G) 3?
Construct one such
MA4235 (Sem 1, 2013/14)
Tutorial 1
1. (a) Let H be a spanning subgraph of a connected graph G. Show that
(H) (G).
(b) For each of the following cases, construct a connected graph G and an
induced subgraph H of G satisfying the inequality:
(i) (H) < (G);
(
MA4235 (Sem 1, 2013/14)
Tutorial 3
1. Let G be the following graph:
(a) Does G have a perfect matching?
(b) Find four maximum matchings in G.
(c) Is there any maximum matching in G that contains the edge cd?
(d) Find four maximal matchings that are not ma
MA4235 (Sem 1, 2013/14)
Tutorial 6
1. Find the values of (G), i(G) and R(G) for the following tree G:
2. Let f = (V0, V1, V2) be a R-function for a graph G of order n 2.
Show that
(i) ([V1]) 1,
(ii) no edge in G joins V1 and V2,
(iii) each vertex in V0 is
MA4235 (Sem 1, 2013/14)
Tutorial 5
1. Evaluate ( (G), (G), (G), (G), (G), i(G) ), where G is the graph
given below.
2. Let G be the graph given in Problem 1 and S = cfw_a, b, u, z. Find the
private neighborhoods pn[a, S], pn[b, S], pn[u, S] and pn[z, S].
YOLUIIU lO RO.I, JUIU 20m
Graphs
and Their
Applications (2)
Koh Khee Meng
Department of Mathematics
National University of Singapore
Singapore 117543
VOLUIIU ~0 00.1, JUJU 20m
Graphs and
Their Applications (2)
5. Connectedness
The mathematical structure:
5. Connectivity
In a connected graph there is at least one path between every pairIfofinits
a vertices.
graph, it happens that by deleting a vertex, or by removing an edge, or performing both,
the graph becomes disconnected,
we can say that such vertices
Chapter 1
Eulerian Multigraphs &
the Chinese Postman Problem
Outline
Eulerian Multigraphs and Semi-Eulerian
Multigraphs
Characterizations
Fleurys Algorithm
Chinese Postman Problem
The Konigsberg Bridge Problem
Bridges
Pregel River
1736, Konigsberg, Pr
3. Sufficient Conditions
Diracs Theorem
What is the least density expected of G to
ensure the existence of a H-cycle in G?
One of the most relevant quantities to
measure how dense a graph is the degree of a
vertex.
Dirac (1952)
Let G be a graph of order
Chapter 3 Connectivity
1. Motivation
Motivation
Cycle
Tree
Wheel
Complete graph
2. The vertex- & edge-connectivity
Assume that G : connected, n = v(G).
A proper subset S of V is called a cut if
G S is disconnected.
A vertex w in G is called a cut-vertex i
Chapter 2
Hamiltonian Graphs &
the Traveling Salesman Problem
1. H-graphs
A connected graph of order n 3
is called a Hamiltonian graph
(H-gp) if it contains a spanning
cycle.
If G is a H-graph, then any
spanning cycle of G is also called
a H-cycle of G.
E
4. The Travelling Salesman
Problem (TSP)
A
7
9
4
B
8
A B C D A 7 8 6 9 30
D
5
C
6
A C B D A 5 8 4 9 26
A B D C A 7 4 6 5 22
Triangle Inequality
w( xy ) w( yz ) w( xz )
b
w( ab) w(bc ) w(ac )
3
2
a
c
7
3 4 6
b
4
3
a
7
2 3 7
triangle inequality not satisfie
Norlan Bonifacio R. De Jesus
SAD
INF 144
Prof. Yabut
1. Use STROBE to compare and contrast Evanss and Ketchams offices. What sort of
conclusion about each persons use of information technology can you draw from your
observations? How compatible do Evans a
BimBO
Sa ganyang baba, siguro pag humihindi ka pwede ka ng makasuntok ng tao. Pag
umu-oo ka naman pwedeng pwede ka ng pampukpok ng pako.
Pangkoy
Tapos, sabi nya unano daw ako. puro height jokes na naman. Hindi ka pa ba
nagsasawa? Kasi kapag ikinumpara yan
Chapter 2
Continuous Time Markov Chains
2.1
Pure Birth Process
In this chapter, we deal with a family of r.v. cfw_X(t); 0 t < where the possible values
of X(t) are nonnegative integers. We shall restrict attention to the case where cfw_X(t)
is a Markov p