D1 - Don't get it.

Draw two triangles. This is still Eulerian, but it's not connected or Hamiltonian, because to get from one triangle to another you'd have to draw a new edge joining them.

2. Eulerian if every vertex has an even number of edges coming from it

Draw a triangle. This is Eulerian, because coming off each vertex there are 2 edges (and 2 is an even number). It is connected, because from each vertex you can get to another vertex by only moving along edges. It's Hamiltonian because you can draw a line through each vertex without going over any edges twice (e.g. just two sides of the triangle).

Draw two triangles. This is still Eulerian, but it's not connected or Hamiltonian, because to get from one triangle to another you'd have to draw a new edge joining them.

Now draw a square with a line going across one of the diagonals. This is still connected, because you can get from any vertex to any other vertex by moving along edges. It's not Eulerian because two of the vertices have 3 lines coming out of them, but it is Hamiltonian because you can draw a path which meets each vertex once and doesn't go over any edge twice (e.g. an N shape or a U shape).

Draw the same square again, but this time with a double-edge along the diagonal. This graph is now connected, Eulerian and Hamiltonian.

Now draw a dot with three dots around it, and connect each outer dot to the inner dot by one edge (i.e. a Y shape). This is connected, but isn't Eulerian (each vertex has an odd number of nodes) and isn't Hamiltonian (to visit each vertex you need to go from the centre to an outer dot and then back again, meaning you overlap an edge).

In summary, a graph is:
1. connected if given any two vertices you can draw a line from one to the other without creating any new edges (you just move along the edges)
2. Eulerian if every vertex has an even number of edges coming from it
3. Hamiltonian if it is connected and you can visit every vertex exactly once without traversing any edges more than once

Last time I checked, each vertex being of even degree was a necessary conditon for an Eulerian graph, but not sufficient. It needs to be connected as well.

Oh, thanks for that, immensely helpful
Is there any way you can check whether a graph is Hamiltonian via a check, rather than actually trying to find whether or not you can reach every node without using a edge thats already been used?

Apparently there are two alternative definitions.

Can you tell me where you've come across the definition that doesn't require the graph to be connected? It's not a variation I've come across before.

Edit: On further checking, I notice that the requirement for connectivity is a technicality to avoid graphs with isolated nodes, but even that definition would not consider two separate triangles to be an Eulerian graph.

I'm not going to go into how much I hate D1, how crap the book is for AQA, and how many times more I'd rather be doing mechanics.

But I just don't seem to get the difference between a connected graph, a Hamiltonian graph, and an Eulerian graph.

Well I get that an Eulerian graph must have vertices of even order. But I seem to have cone across 85473 conflicting definitions & examples of what a connected and Hamiltonian graph is. If anyone could help, it'd be hugely appreciated.

please do
are you also using the oxford press books, the ones which are full of mistakes and examples completely unrelated to the questions?
And yeah, don't want to rub it in but if you do physics at AS and M1 at A2 it's a godsend.

Go over the past papers with the mark scheme and use the book to help you if you don't understand the answers..