Solved on Dec 06, 2023
A connected graph with even vertices and odd vertices has an Euler circuit.
STEP 1
Assumptions
1. A connected graph is given.
2. The graph has 60 vertices with even degrees.
3. The graph has 2 vertices with odd degrees.
STEP 2
Recall the necessary conditions for a graph to have an Euler path and an Euler circuit.
- A connected graph has an Euler circuit if and only if every vertex has an even degree.
- A connected graph has an Euler path but not an Euler circuit if and only if exactly two vertices have an odd degree.
STEP 3
Since the graph has 60 vertices with even degrees, these vertices satisfy the condition for both an Euler path and an Euler circuit.
STEP 4
However, the graph also has 2 vertices with odd degrees. This violates the condition for an Euler circuit, which requires all vertices to have even degrees.
STEP 5
The presence of exactly two vertices with odd degrees satisfies the condition for the existence of an Euler path.
STEP 6
Because the graph is connected and has exactly two vertices with odd degrees, we can conclude that the graph has an Euler path but not an Euler circuit.
The described graph has an Euler path (but not an Euler circuit).
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