The document provides examples and definitions related to graph theory terminology. It defines types of graphs such as directed graphs, multigraphs, weighted graphs, and bipartite graphs. It also provides examples to illustrate concepts like coloring cubes, crossing a river with constraints, and properties of connected components in an undirected graph.

1. 7.1. Preliminaries &
Graph Terminology
Reference: A Course in Discrete Mathematical
Structures by LR Vermani
Budi Irmawati-NAIST’s student

2. 2
Isomorphic
G (V,E) G’(V’,E’)
x (x,y) α=VV’ x’ =α(x)
y β=EE’ y’ =α(y) (x’,y’) = β(x,y)
z z’ =α(z)
4 3 d
c
c
d
1 2 a b a b

3. 3
Multigraph
Graph that has multiple
edges connecting the same
vertices
Directed Multigraph
Directed graph that is
multigraph
Simple/Linear Graph
graph which no loop and only
have one edge between two
vertices

4. 4
Weighted graph
Graph which each of its
edges are weighted.
Ordered quadruple (V,E,f,g)
or triple (V,E,f) or triple (V,E,f)
V : non-empty set of vertices
E : set/multiset of edges
f : function with domain V
g : map with domain E

5. 5
Bipartite
V
Graph V1 V2 sub graph 1
(male) (female) Each vertex in V1 incident
with an vertex in V2
a1 b1
a2 b2
⋮ ⋮
an bn
V sub graph 2
Every vertex in V1 may not
V1 V2 incident with a vertex in V2
(male) (female)
Some vertex in V1 or V2 are
a1 b1 isolated
disjoint a2 b2 disjoint
⋮ ⋮
an bn

6. 6
Bipartite
Graph Fig. 7.33
V1 = {a,b,g}
V2 = {c,d,e,f}
Fig. 7.34
V = V1 V2
V1 V2 =
V1
V2
not bipartite

7. 7
Bipartite
Complete Bipartite
Graph Graph
V = V1 V2
V1 V2 =
V1
V2
Every vertex in V1 adjacent
every vertex in V2 and vice
versa
Km,n
Complete bipartite graph
which V1 order m and V2
order n
The number of edges is mn
instead of n(n-1)/2
Every vertex adjacent to
every vertex in other
subgraph

8. 8
Bipartite Graph
simple graph: two vertices – one edge simple graph: at least three vertices (possibly)
 always bipartite  trivial graph; three isolated
vertices
 only have one edge
always
 have two edges, or bipartite
 have three edges

9. 9
Example 7.2
Cube with 6 faces. Every cube are colored using 4 colors.
Is it possible to stack the cubes to form column so that no color
apears twice

10. 10
1 2 3 4
a R G W B
b G W B R
c W R B G
Example 7.2
d G B W R Cube with 6 faces. Every cube are colored using 4 colors.
Is it possible to stack the cubes to form column so that no color
apears twice

11. 11
Possible
crossing
one person
a couple
three ladies
three gents
couple & gent
Example 7.3
How to cross a river for:
5 couples
1 boat for 3 persons
wife + husband if there a man

12. 12
Example 7.4
How to cross a river for:
5 couples
1 boat for at most 4 persons
wife + husband if there a man

13. 13
Example 7.5
A man, a dog, a sheep, a basket of cabbage
Only can carry one item in crossing a river
Cabbage cannot stay with a sheep and a dog cannot stay with a sheep
How to cross it ?

14. 14
Example 7.6
G = (V,E) is undirected graph with k components
o(V) = n, o(E) = m
Prove that m ≥ n - k
 G’ = (V’,E’) where o(V’) = s ≥ 2.
Vertex v1. G’ is connected graph.
 v1 adjacent to v2, v2 to v3, and so on.
Chain v1v2, v2v3, …, vs-1vs.
 at least s-1 edges in G’
For o(V’) = 1, o(E’) ≥ o(V’) – 1

15. 15
Example 7.6
 For k components, (E1,V1), (E2,V2), …, (Ek,Vk)
 Ei Ej = for i j
 E = E1 E2 … Ek and V = V1 V2 … Vk
 o(E) = o(E1)+o(E2)+…+o(Ek)
≥ (o(V1)-1)+(o(V2)-1)+…+(o(Vk)-1))
= o(v1)+o(V2)+…+o(Vk) – k
= o(V) - k