The degree sequence of a simple graph is the sequence of the degrees of the nodes in the graph in decreasing order. Which of the following sequences can not be the degree sequence of any graph? I. 7,6,5,4,4,3,2,1 II. 6,6,6,6,3,3,2,2 III. 7,6,6,4,4,3,2,2 IV. 8,7,7,6,4,2,1,1 A. I and II B. III and IV C. IV only D. II and IV

GATE 2010 · Discrete Mathematics · Degree of Graph · medium

Answer: II and IV are not graphical (Option D)

1. Max-degree check (n = 8): IV begins 8,... but max allowed degree is n-1 = 7, so IV is impossible immediately
2. Havel-Hakimi on II: 6,6,6,6,3,3,2,2: 6:6,6,6,3,3,2,2->5,5,5,2,2,1,1; 5:5,5,2,2,1,1->4,4,1,1,0,1=4,4,1,1,1,0; 4:4,1,1,1,0->3,0,0,0,0; remove 3 needs three positive entries but only zeros remain -> negative -> NOT graphical
3. Havel-Hakimi on I and III (both pass): I: 7,6,5,4,4,3,2,1 reduces cleanly to all zeros; III: 7,6,6,4,4,3,2,2 reduces cleanly to all zeros -> both graphical