What are the best options that allow running a Linux distribution, such as Ubuntu inside Windows XP, i.e. within a window?

By "best" I mean quick startup time, low memory requirement, good graphical performance, easy access to the main (host) Windows XP file system.

I know about Wubi and standard Ubuntu with the dual boot capability, but I'd prefer to be able to switch from XP to Linux without a reboot.  I know the Linux experience may not be ideal this way, but it would be pefectly good for my needs.

I've downloaded VirtualBox and the latest Ubuntu and will try the virtual machine approach; however are there any specifically customised Linux versions for this purpose?

I think this would be a great way for people new to Linux to get used to the OS while still having the "safety net" of Windows.


Question: Cubic graphs without a perfect matching and a vertex incident to three bridges
Answer: Substitute your central vertex in your graph with a 3-cycle $abc$ so that the graph stays cubic. Now subdivide each edge in this 3-cycle. So we have new vertices $u$ connected to $a$ and $b$, $v$ connected to $b$ and $c$, $w$ connected to $c$ and $a$. Now add a final vertex $x$ and connect it to $u,v$ and $w$. This graph has exactly three bridges, none of which intersect the other at a vertex, and moreover has no perfect matching!

One result which relates the existence of a perfect matching in a cubic graph and its bridges is the following theorem of Petersen from "Die theorie der regularen graphen", Acta Math. 15
(1891), 163-220:


  Theorem: Every cubic graph with at most two bridges contains a perfect matching. 


As well as this strengthening by Errera, "Du colorage des cartes", Mathesis 36 (1922), 56-60:


  Theorem: If all the bridges of a connected cubic graph $G$ lie on a single path of $G$, then $G$ has a perfect matching.


So your instinct is true, in the sense that if the graph has no perfect matching, its bridges do not lie on a path. However the example in the beginning of this answer shows that they are not necessarily incident at the same vertex.