Perhaps slightly off with respect to usefulness but certainly accessible : The four color theorem. (I am a bit surprised that nobody mentioned it, if I am mistaken I should change my glasses). It has a fascinating history and the use of computers in its proof is still slightly controversial. This would allow to discuss the notion of 'proof' (the first proof was riddled with errors but was nevertheless more or less accepted since all errors could easily be corrected). It has interesting reformulations (for example in terms of cross-products) and can be generalized to graphs embedded in other surfaces. It can also be complemented for example by mentioning the (still unknown exact) value of the chromatic number of the Euclidean plane (minimal number of colors needed to color all points of $\mathbb R^2$ with no points of identical color at distance $1$) or generalizations to other surfaces. One could also end with a Theorem of Steinitz (a graph is the $1$-skeleton of a polytop if and only if it is planar and can not be disconnected by removing less than $3$ edges), mention the five platonic solids (and perhaps even Schlaefli's classification of regular polytops in higher dimensions).