We start by noticing that four of the circles have three degrees of connectivity and two of them have four degrees of connectivity.
First we observe that if we do not consider number 1 and 6 adjacent then 1 and 6 are each not adjacent to four numbers while 2,3,4,and 5 are each not adjacent to three numbers. This is seen in the adjacency table:
1: 3,4,5,6
2: 4,5,6
3: 1,5,6
4: 1,2,6
5: 1,2,3
6: 1,2,3,4
So this lets us position numbers 1 and 6 as shown in picture A. Since 5 can not be adjacent to 6 and 2 can not be adjacent to 1 the set of numbers positioned in those adjacent circles can only be {1,2,3,4,5,6}-{1,6}-{2,5}={3,4}.
From here we see there are two solutions that solve the puzzle as stated.