Question

2. For each of the following, draw a (simple) graph with the corresponding degree sequence, or explain why no such graph exists. (a) A graph with degree sequence 1, 1, 1, 1. (b) A graph with degree sequence 3, 3, 2, 2, 1, 1, 1. (c) A graph with degree sequence 4, 4, 4, 4, 4, 4. (d) A graph with degree sequence 6, 5, 4, 3, 2, 1

Answer 1

for existence for the simple graph to existing, we can use erdos-Gallai theorem this is if and only if the condition so no worries

Erdös-Gallaithm", A sequence of non-negative Integas diadz...zdu can be represented as the degree sequence of a finite simple graph on nuestices it Edo is then and Ido sklk-1) + Ź mendi, k) holde ferikan. 12K+ hence In (b) f(d) {di isn't even we msn't proceed further (Q = 1,1,1,1 Kr! + ? 161-1) + (1+1+1) frue 1+1% 212-1) + (1+1}xys true R=3 HAIE 3(3-1) + 1 true hence graph exista K=2 K=/ d(C) 4,4,4,4,4,4 di is onen 45 4(1-1) + Š minldi,K) = 0+ (1+1+1+1+1) tove 4+4 5 212-3+ (2+2+2+2) tone 4+4+4 53(3-4) + (3+3+3) fone I hence K=4 graph Кs. 4444444 € 414-1) +14+4) tome ca fecitue! K=6 - 4+4+4+4+42515-1)+ 5 bine 4+4+4+4+4+4 86(619+ true