# euler & hamilton by piyawat.87

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• pg 1
```									Paths & Circuits
Paths & Cycles
Euler Paths
• An Euler path in a graph is a simple path
containing every edges of that graph.
Euler Circuits
• An Euler circuit in a graph is a simple circuit
containing every edges of that graph.
Euler Paths & Circuits
• An Euler circuit in a graph is a circuit that traverses all
the edges in the graph once.
• An Euler path in a graph is a path that traverses all
the edges in the graph once.
• An undirected multi-graph has an Euler circuit if and
only if it is connected and has all vertices of even
degree.
• An undirected multi-graph has an Euler path, but not
Euler circuit, if and only if it is connected and has
exactly two vertices of odd degree.
Condition of Euler Circuits
• A connected multi-graph with at least two
vertices has an Euler circuit ↔ each of its
vertices has even degree.

• Necessary condition: G has an Euler Circuit →
each of its vertices must have even degree.

• Sufficient condition: Each of the vertices in G
has even degree → G has an Euler Circuit.
Find an Euler Circuit
• G has an Euler Circuit → each of its vertices must
have even degree.
• Each of the vertices in G has even degree → G has an
Euler Circuit.
Find an Euler Circuit
Condition of Euler path
• A connected multi-graph with at least two
vertices has an Euler path ↔ it has exactly 2
vertices with odd degree.
Find an Euler Path
Euler Circuit Problem
ลากเส้ นตามรูปโดยไม่ ซ้า

(Euler path)        (Euler Circuit)
Euler Paths & Circuits
Hamilton Paths & Circuits
• A Hamilton path in a graph is a simple path
that passes through every vertex of the graph
exactly once.

• For G=(V,E) and V = {v1, v2, …, vn}, the simple
circuit v1,v2,…,vn,v0 is a Hamilton circuit if
v1,v2,…,vn is a Hamilton path.
Hamilton Paths & Circuits
• A Hamilton circuit in a graph is a (simple) circuit that
visits each vertex in the graph once.
• A Hamilton path in a graph is a (simple) path that
visits each vertex in the graph once
Iconion Puzzle
Condition of Hamilton Circuits
• No ‘necessary & sufficient’ conditions exist.
• Certain properties can be used to show that no
Hamilton circuits exist. E.g. degree one vertex.
• Both edges incident of a vertex of degree two
must be part of any Hamilton circuit.
• While constructing a Hamilton circuit, if a vertex
has already passes through, all remaining edges
of that vertex can be removed from
consideration.
Some sufficient condition
• If G is a simple graph with n vertices (n≥3)
such that the degree of every vertex in G is at
least n/2, then G has a Hamilton circuit.

• If G is a simple graph with n vertices (n≥3)
such that deg(u)+deg(v) ≥ n for every pair of
non-adjacent vertices u and v in G, then G has
a Hamilton circuit.
Hamilton Circuit
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1

5                                       10
2
1        2       3
4            3
13     4            5   6
กราฟที่มีวงจร Hamilton                                    11
7        8       9

12

William Rowan Hamilton (1805–1865)                  กราฟที่ไม่มีวงจร Hamilton
Traveling Salesperson Problem
• A salesman is required to visit a number of cities
during a trip. Given the distances between cities, in
what order should he travel so as to visit every city
precisely once and return home, with the minimum
mileage traveled ?

Input                     Output

```
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