△-free→lEl≤n2/4 Second Proof, ∑(d+d)=d uw∈E u∈V △-free→du+d≤n→∑(du+d)≤nlEl uw∈E Cauchy-Schwarz 2 4E2 (handshaking） m ∈V mE≥∑d+d)=∑e≥②ev4- 4E2 m uU∈E w∈V m
-free 㱺 |E| ≤ n2/4 Second Proof. uvE (du + dv) = vV d2 v u v (du + dv) Cauchy-Schwarz ⇤ vV d2 v 1 n ⇤ vV dv ⇥2 = 4|E| 2 n -free ⇥ du + dv n ⇥ uvE (du + dv) n|E| n|E| ⌅ uvE (du + dv) = ⌅ vV d2 v ⇤ vV dv ⇥2 n = 4|E| 2 n (handshaking)
△-free→lEl≤n214 Second Proof, >(d+d)=d uw∈E v∈V △-free→du+d≤n→∑(du+d)≤nlEl uu∈E Cauchy-Schwarz 2 ≥(4 4纠E2 (handshaking） m v∈V v∈V nE≥ 4到E2 E≤ 2 m
-free 㱺 |E| ≤ n2/4 Second Proof. uvE (du + dv) = vV d2 v Cauchy-Schwarz ⇤ vV d2 v 1 n ⇤ vV dv ⇥2 = 4|E| 2 n -free ⇥ du + dv n ⇥ uvE (du + dv) n|E| n|E| 4|E| 2 n (handshaking) |E| n2 4 u v (du + dv)
△-free→lEl≤n2/4 Third Proof. A:maximum independent set a lAl independent u∈V,d,≤a dv B=V\A B incident to all edges B=IBI >0 Inequality of the arithmetic and geometric mean B s∑4≤a9s()- 4 u∈B
-free 㱺 |E| ≤ n2/4 Third Proof. A: maximum independent set B = V \ A α = |A| β = |B| v ⇤ ⇥ ⌅ dv independent ⇤v ⇥ V, dv B B incident to all edges ⇥ + ⇥ 2 ⇥2 |E| vB dv = n2 4 Inequality of the arithmetic and geometric mean
Turan's Theorem "Suppose G is a Kr-free graph. What is the largest number of edges that G can have?" Paul Turan (1910-1976)
Turán's Theorem Paul Turán (1910-1976) “Suppose G is a Kr -free graph. What is the largest number of edges that G can have?
Turan's Theorem Theorem (Turan 1941) If G(V.E)has IV=n and is K,-free,then E卧≤ r-2n2 (r-1)
Turán's Theorem Theorem (Turán 1941) If G(V,E) has |V|=n and is Kr-free, then |E| ⇥ r 2 2(r 1)n2