There is an easy way to check if a graph is traversable by using the degrees of the vertices.
Key point 1.15 An Eulerian graph is a connected graph that has no vertices of odd degree. Eulerian graphs are traversable, with the trail starting and finishing at the same vertex. A semi-Eulerian graph is a connected graph that has exactly 2 vertices of odd degree. Semi-Eulerian graphs are traversable, but the trail starts at one of the odd vertices and finishes at the other odd vertex.
WORKED EXAMPLE 1.16
PL E
It is easy to show that each time a trail enters and exits a vertex it uses up 2 edges, so for a graph to be traversable the degrees must all be even, apart from the start and finish which will have odd degree (or even if the trail finishes where it started). However, it is much more difficult to prove that every connected graph with 0 or exactly 2 vertices of odd degree is traversable.
A simple-connected graph has 4 vertices and 5 edges. The vertex degrees are 2, 3, x and y. a Show that the graph must be semi-Eulerian.
b Explain why the missing degrees must be 2 and 3.
a
M
Drawing an example of a graph that fits the description is not enough, because there might be more than one possibility.
2 + 3 + x + y = 10
so
x + y = 5
The sum of the degrees is even.
is odd
SA
x + y
The sum of the degrees is twice the number of edges.
One of the missing degrees is odd and the other is even.
The graph has exactly 2
odd degrees, so it must be semi-Eulerian.
Odd + even = odd Odd + odd or even + even = even Exactly 2 odd degrees ⇒ semi-Eulerian
b Simple-connected: x, y ∈ {1, 2, 3}
Connected: x, y
x + y = 5
so x
= 2, y = 3
⩾ 1
. Simple: x, y
or
.
x = 3, y = 2
Original material © Cambridge University Press 2018
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