proofs

Turán’s graph theorem 239
Case 2: d(u) ≥ d(v) and d(u) ≥ d(w).
Duplicate u twice and delete v and w (as illustrated in the margin). Again,
the new graph G ′ has no p-clique, and we compute (the −1 results from the
edge vw):
v
u
w
u ′
u ′′
|E(G ′ )|
= |E(G)| + 2d(u) − (d(v) + d(w) − 1) > |E(G)|.
So we have a contradiction once more.
A moment’s thought shows that the claim we have proved is equivalent to
the statement that
u ∼ v :⇐⇒ uv ∉ E(G)
defines an equivalence relation. Thus G is a complete multipartite graph,
G = K n1,...,n p−1
, and we are finished.
□
References
[1] M. AIGNER: Turán’s graph theorem, Amer. Math. Monthly 102 (1995),
808-816.
[2] N. ALON & J. SPENCER: The Probabilistic Method, Wiley Interscience 1992.
[3] P. ERDŐS: On the graph theorem of Turán (in Hungarian), Math. Fiz. Lapok
21 (1970), 249-251.
[4] T. S. MOTZKIN & E. G. STRAUS: Maxima for graphs and a new proof of a
theorem of Turán, Canad. J. Math. 17 (1965), 533-540.
[5] P. TURÁN: On an extremal problem in graph theory, Math. Fiz. Lapok 48
(1941), 436-452.
“Larger weights to move”