# Hamiltonian and Eulerian graph with 2 bridges?

Is it possible to have graph which is Hamiltonian and Eulerian at once with 2 bridges in it?

If yes, how can it look like?

Thank you.

Edited:

The bridge is an edge of a graph whose deletion increases its number of connected components.

Graph with one bridge - Bridge here is edge between vertices 3 and 4.

• Remind me what a "bridge" means in this context? – Thomas Andrews May 21 '19 at 20:14

A connected graph having a bridge is not Eulerian. Let the bridge be $$e$$, and the connected components that result on deletion of $$e$$ be $$C_1,C_2$$. Notably, $$e$$ is the only edge with one end in $$C_1$$ and the other in $$C_2$$. The Euler line, if it exists, may begin at some vertex in $$C_1$$ and must contain $$e$$. When $$e$$ is included in the walk, we enter $$C_2$$. Since the Euler line must terminate at the same vertex in $$C_1$$, we must re-enter $$C_1$$. But this is not possible, as the only edge with its ends in different components is $$e$$, which cannot be repeated in the walk.