For each of the graphs shown below, first give an unique name to each edge (e.g.

Question

For each of the graphs shown below, first give an unique name to each edge (e.g., a, b ... etc.), and then determine if a graph has Euler path or not, as well as a Hamiltonian cycle or not. If a graph does not have an Euler path, say "No". If a graph has a Euler path, please list the path using the similar format shown in the previous example: node name, edge name node name, edge name, , , , node name. If a graph does not have an Hamiltonian cycle, say "No". If a graph has a Hamiltonian cycle, please list the path using the similar format shown in the previous example: node name, edge name, node name, edge name, , , , node name.

Explanation / Answer

Euler Graph:-

A graph G is euler graph if and oly if it is connected and all vertices have even degree.

A graph G is euler graph if and only if it have euler cycle.

A graph G has euler path if and only if it is connected and all vertices have even degree except 2 vertices one is start and end vertices.

so graph

d) it is connected but all vertices does not have even degree so it is not euler graph .so it does not have euler cycle.

and it does not have eulerr path because more then 2 vertices have odd degree.like vertex 1,3,5,7

e) it is connected but all vertices does not have even degree so it is not euler graph .so it does not have euler cycle.

and it does not have eulerr path because more then 2 vertices have odd degree.like vertex 1,9,5,6

f) it is connected but all vertices does not have even degree so it is not euler graph .so it does not have euler cycle.

and it does not have eulerr path because all vertices have odd degree.

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For hamaltoniaon cycle:-

d and e does not have haalotinon cycle.

f ) it has hamiltonian cycle

1 4 5 6 7 3 2 1

first we go 1 to 4 then 4 to 5 then 5 to 6 then 6 to 7 then 7 to 3 thne 3 to 2 then 2 to 1.

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