2. topics
• Definition of graph
• Some important points
• Types of graphs
• walk,path & trail
• hamiltonian path & circuit
• Euler path & circuit
• Colouring of graph
3. Definition of graph
• Formally, a graph is a pair of sets (V,E), where V
is the set of vertices and E is the set ofset of
• edges, formed by pairs of vertices
4. Some important points
• Loop and Multiple Edges
• A loop is an edge whose endpoints are equal i.e.,
an edge joining a vertex to it self is called a loop.
We say that the graph has multiple edges if in the
graph two or more edges joining the same pair of
vertices.
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5. Types of graph-
Simple Graph
A graph with no loops or multiple edges is called a simple graph. We
specify a simple graph by its set of vertices and set of edges,
treating the edge set as a set of unordered pairs of vertices and
write e = uv (or e = vu) for an edge e with endpoints u and v.
Connected Graph
A graph that is in one piece is said to be connected, whereas one
which splits into several pieces is disconnected.
6. • Subgraph
• Let G be a graph with vertex set V(G) and edge-list E(G). A
subgraph of G is a graph all of whose vertices belong to V(G)
and all of whose edges belong to E(G). For example, if G is
the connected graph below:
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w, z} and E(G) = (uv, uw, vv, vw, wz, wz} then the following four graphs ar
7. • Degree (or Valency)
• Let G be a graph with loops, and let v be a vertex of G. The degree of v is
the number of edges meeting at v, and is denoted by deg(v).
• For example, consider, the following graph G
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The graph G has deg(u) = 2, deg(v) = 3, deg(w) = 4 and deg(z) = 1.
Regular Graph
A graph is regular if all the vertices of G have the same degree. In particular, if
the degree of each vertex is r, the G is regular of degree r.
8. • Isomorphic Graphs
• Two graph G and H are isomorphic if H can be obtained from G
by relabeling the vertices - that is, if there is a one-to-one
correspondence between the vertices of G and those of H, such
that the number of edges joining any pair of vertices in G is
equal to the number of edges joining the corresponding pair of
vertices in H. For example, two labeled graphs, such as
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9. • Walk
• A walk of length k in a graph G is a succession of k edges of
G of the form uv, vw, wx, . . . , yz.
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We denote this walk by uvwx . . yz and refer to it as a walk between u and z.
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Trail and Path
• all the edges (but no necessarily all the vertices) of a walk are different, then
If
the walk is called a trail. If, in addition, all the vertices are difficult, then the trail
is called path.
The walk vzzywxy is a trail since the vertices y and z both occur twice.
The walk vwxyz is a path since the walk has no repeated vertices.
10. Hamiltonian path & circuit
• a Hamilton path in the graph (named after an
• Irish mathematician, Sir William Rowan Hamilton)., a
Hamilton path is a path that visits every vertex in the
graph
• A Hamilton circuit is a path that visits every vertex in the
graph exactly
• once and return to the starting vertex.
a b
a b
d c d c
11. Euler path & circuit-
Euler Path is a path in the graph that passes
each edge only
once.
Euler Circut is a path in
the graph that passes
each edge
only once and return back
to its original position.
From Denition, Euler
Circuit is a subset of Euler
Path
12. Colouring of graph
Vertex Coloring
Let G be a graph with no loops. A k-coloring of G is an assignment of k colors
to the vertices of G in such a way that adjacent vertices are assigned different
color
4-coloring 3-coloring
It is easy to see from above examples that chromatic number of G is
at least 3. That is X(G) ≤ 3, since G has a 3-coloring in first diagram.
On the other hand, X(G) ≥ 3, since G contains three mutually
adjacent vertices (forming a triangle)., which must be assigned
different colors. Therefore, we have X(G) = 3.
13. Edge Colorings
Let G be a graph with no loops. A k-edge-coloring of G is an
assignment of k colors to the edges of G in such a way that any
two edges meeting at a common vertex are assigned different
colors,
5-edge-coloring
4-edges-coloring
From the above examples, it follows that X`(G) ≤ 4, since G has a 4-edge-
coloring in figure a (above). On the other hand, X`(G) ≥ 4, since G
contains 4 edges meeting at a common vertex i.e., a vertex of degree 4,
which must be assigned different colors. Therefore, X`(G) = 4.
