Problem

Source: Mongolia TST 2011 Test 3 #3

Tags: function, graph theory, combinatorics unsolved, combinatorics



Let $n$ and $d$ be positive integers satisfying $d<\dfrac{n}{2}$. There are $n$ boys and $n$ girls in a school. Each boy has at most $d$ girlfriends and each girl has at most $d$ boyfriends. Prove that one can introduce some of them to make each boy have exactly $2d$ girlfriends and each girl have exactly $2d$ boyfriends. (I think we assume if a girl has a boyfriend, she is his girlfriend as well and vice versa) (proposed by B. Batbaysgalan, folklore).