Is there a graph that has 7 vertices and each vertex has a degree of $2,2,3,5,5,5,6$? Is there a graph that has 7 vertices and each vertex has a degree of $2,2,3,5,5,5,6$?
Any ideas on how to solve this one?
 A: Vertex #7 must be joined to each of the others.  This leaves degrees 1,1,2,4,4,4,0 free.  At least one of vertices #4, #5, #6 won't be joined to 
vertex #3.  WLOG this is #6.  It must be joined to #1, #2, #4, and #5.  This leaves 0, 0, 2, 3, 3, 0, 0.  But there are no 3 more vertices to join #5 to.    
A: According to Havel-Hakimi theorem for possible degree sequence in the graph.
You given degree sequence $2, 2, 3, 5, 5, 5, 6$ 
Step$(1):$ Sort the given sequence in decreasing order : $6, 5, 5, 5, 3, 2, 2$ 
Step$(2):$ Remove degree sequence $6$ and subtract one from only six degree sequence  : $ 4, 4, 4, 2, 1, 1$ 
Step$(3):$ Sort the given sequence in decreasing order : $ 4, 4, 4, 2, 1, 1$ (already sorted). 
Step$(4):$ Remove degree sequence $4$ and subtract one from only four degree sequence  : $ 3, 3, 1, 0, 1$ 
Step$(5):$ Sort the given sequence in decreasing order : $ 3, 3, 1, 1, 0$ 
Step$(6):$ Remove degree sequence $3$ and subtract one from only three degree sequence  : $ 2, 0, 0, 0$ 
Step$(7):$ Sort the given sequence in decreasing order : $ 2, 0, 0, 0$ (already sorted).  
Step$(8):$ Remove degree sequence $2$ and subtract one from only two degree sequence  :$-1, -1, 0$ 
Step$(9):$ Stop, since degree sequence of a graph can not be negative, so your given degree sequence is not possible of a graph,i.e. invalid degree sequence.

Note: when resultant degree sequence has total number of $1's$ is even, then degree sequence of a graph is possible.  
