A 3-bit string is a string made up of 3 characters, each a 0 or a 1. Suppose youd like to write out, in one string, all eight of the 3-bit strings in any convenient order.

For example, if you wrote out the 3-bit strings in the usual order starting with 000, 001, 010, . . . , you could concatenate them together to get a length 3 times 8 = 24 24-string that started 000001010. . . .

But you can get a shorter string containing all eight 3-bit strings by starting with 00010. . . . In which Now 000 is present as bits 1 through 3, and 001 is present as bits 2 through 4, and 010 is present as bits 3 through 5, . . . .

1. The situation above generalizes to k >=2. Namely, for each k there is a digraph, B k, such that the nodes are V(B k)={ 0 , 1 } k, and any walk through B k that contains every edge exactly once determines a minimum length , (k+1)-kool bit-string. What is this minimum length?
2. Define the transitions of Bk. Verify that the in-degree and out-degree of every vertex is even, and that there is a positive path from any vertex to any other vertex (including itself) of length at most k.

+1 10 +0 110 +1 +0 +1 +0 100 +0. +1