I am just learning for my exam of Graph Theory in the University and I found in my notes the definition of map:
There are in the notes from lectures pictures of map explaining why the definition of map is what it is. And, if it is connected with classical definition of this word, I started wondering why there can't be bridges. For instance, let's consider connection of Spain and Portugal. Isn't it a bridge? If we removed their boundary, we would get the two graphs: Europe without Portugal and Portugal without Europe. Is my definition or understanding wrong?
The formal graph definition of a map and our real world notion of map are slightly different.
The real maps contain regions which are not connected (for example Azerbaijan). And this becomes even more complex if you consider the oceans/seas as a single region.
The graph theory definition is of an "ideal" map, not any map. Also remember that a map is a graph drawn on the entire plane/sphere, there is no end to it. The border between Spain and Portugal is not a bridge, you have to consider the ocean as a region. [this would correspond to the infinite face in the graph, if you consider the map as being "planar".] And the boundary between Portugal and the ocean is a second edge connecting Spain and Portugal, none is a bridge.
A bridge would be a border between a country and itself. This is not possible on a real map either.
And the faces of a graph having a bridge cannot be colored, the bridge separates a region and itself, so the region should be colored with two different colors.
This is easier to understand looking at the dual graph. Any bridge becomes a loop in the planar graph, and you cannot color the vertices of a graph with a loop.