I read a book about American English. It reports that, in standard informal conversations, American English doesn't use the /ɔ/ sound; it uses the /ɑ/ sound and /ʌ/ and /ə/ are not different. Are they really?
That book would not use the /ɔ/ and /ʌ/ sounds, but when I look in my American English Dictionary for some words, such as more, door, and love, they are reported to be pronounced /mɔr/, /dɔr/, and /lʌv/.
How should I pronounce these words, if there are no /ɔ/and /ʌ/ sounds?  Should they be /mɑr/, /dɑr/, and /ləv/?
Can /ɑ/ sound replace /ɔ/, and /ə/ replace /ʌ/ in every word?
What about formal American English? Does it have /ɔ /and /ʌ/ sounds or not?


Question: Question: Graph Theory and Trees
Answer: By a theorem of Ore (proof here), if every pair of non-adjacent vertices of a graph on $m$ vertices has degrees adding up to at least $m$, then the graph is Hamiltonian. This applies to the schoolchildren, with $m=2n$. So they can be ordered into a Hamiltonian cycle (a path will do), $c_1,c_2,\dots,c_{2n}$, where each $c_i$ is friends with $c_{i-1}$ and $c_{i+1}$. Now pair $c_1$ with $c_2$, $c_3$ with $c_4$, and so on. 

Dirac's Theorem, a special case of Ore's Theorem, is strong enough to solve this problem. A proof is here. 

EDIT: Dirac's Theorem states that if each of the $m$ vertices of a graph has degree at least $m/2$, then the graph is Hamiltonian. This is exactly the hypothesis satisfied by the schoolchild graph.