Explicit Construction of Small Folkman Graphs

Transcription

1 Michael Spectral Graph Theory Final Presentation April 17, 2017

2 Notation Rado s Arrow Consider two graphs G and H. Then G (H) p is the statement that if the edges of G are p-colored, then there exists a monochromatic subgraph of G isomorphic to H. K 6 (K 3 ) 2 K 5 (K 3 ) 2

3 Conjecture: Erdős and Hajnal (1967) For each p there exists a graph G, containing no K 4, which has the property that G (K 3 ) p. The case when p = 2 was proven by the following theorem: Folkman s Theorem (1970) For any k 2 > k 1 3, there exist a K k2 -free graph G with G (K k1 ) 2. Thus, any K 4 -free graph G with G (K 3 ) 2 is a Folkman Graph.

4 Conjecture: Erdős and Hajnal (1967) For each p there exists a graph G, containing no K 4, which has the property that G (K 3 ) p. Folkman s theorem was generalized as: Ne set ril - Rödl s Theorem (1976) For p 2 and any k 2 > k 1 3, there exist a K k2 -free graph G with G (K k1 ) p. Thus, there exist a K 4 -free graph G with G (K 3 ) p.

5 Question For any k 1 < k 2 and any p 2, what is the smallest integer f (p, k 1, k 2 ) = n such that there is a K k2 -free graph G on n vertices satisfying G (K k1 ) p. Graham (1968) proved that f (2, 3, 6) = 8 by showing that K 8 \ C 5 (K 3 ) 2.

6 Question For any k 1 < k 2 and any p 2, what is the smallest integer f (p, k 1, k 2 ) = n such that there is a K k2 -free graph G on n vertices satisfying G (K k1 ) p. Graham (1968) proved that f (2, 3, 6) = 8 by showing that K 8 \ C 5 (K 3 ) 2.

7 Question For any k 1 < k 2 and any p 2, what is the smallest integer f (p, k 1, k 2 ) = n such that there is a K k2 -free graph G on n vertices satisfying G (K k1 ) p. Graham (1968) proved that f (2, 3, 6) = 8 by showing that K 8 \ C 5 (K 3 ) 2.

8 Question For any k 1 < k 2 and any p 2, what is the smallest integer f (p, k 1, k 2 ) = n such that there is a K k2 -free graph G on n vertices satisfying G (K k1 ) p. Graham (1968) proved that f (2, 3, 6) = 8 by showing that K 8 \ C 5 (K 3 ) 2.

9 Question For any k 1 < k 2 and any p 2, what is the smallest integer f (p, k 1, k 2 ) = n such that there is a K k2 -free graph G on n vertices satisfying G (K k1 ) p. Graham (1968) proved that f (2, 3, 6) = 8 by showing that K 8 \ C 5 (K 3 ) 2.

10 K 5 -free graphs G with G (K 3 ) 2 : n = f(2,3,5) Authors n Graham, Spencer (1971) 23 Irving (1973) 18 Khadzhiivanov and Nenov (1979) 16 Nenov (1981) 15 Piwakowski, Radziszowski, and Urbański (1998) 15

11 Bounds for f(2,3,4) Both upper bounds of Folkman and of Ne set ril and Rödl for f (2, 3, 4) are extremely large. Frankl and Rödl (1986) showed f (2, 3, 4) Erdös $$ prize Erdös set a prize of $100 for the challenge f (2, 3, 4) Spencer (1988) showed that f (2, 3, 4) Again, Erdös set a prize of $100 for the challenge f (2, 3, 4) This paper claimed the reward in 2008!

12 $100 Theorem Theorem: LU (2007) f (2, 3, 4) Proof Sketch Use spectral analysis to establish a sufficient condition for G (K 3 ) 2. Examine a special class of graphs and find the four small Folkman graphs of size 9697, 30193, 33121, Get that cash money.

13 The Graph L(m, s) Let gcd(m, s) = 1 and let n be the smallest integer such that L(m, s) s n 1 mod m. The graph L(m, s) is the circulant graph on m vertices generated by S = {s i mod m : i = 0, 1,..., n 1}. Spectrum Of Circulant Graphs The eigenvalues of the adjacency matrix for the circulant graph generated by S Z n are ( ) 2πis cos, n for i = 0, 1,..., n 1. s S

14 The Graph L(m, s) Lemma H, the unique local graph of L(m, s) is isomorphic to a circulant graph of order n. proof sketch: V (H) = S, E(H) = {xy : x S, y S, x y S}. Define the bijection f : Z n S such that f (i) = s i mod n. Since f (i + j) = f (i)f (j), f is a group isomorphism. Define T Z n such that T = {i : f (i) 1 S}. Let H be the circulant graph generated by (Z n, T ). f is in fact a group homomorphism, mapping H to H.

15 Proof Sketch Of Main Theorem H is a local graph of L(m, s) A is the adjacency matrix for H. Let σ = λ min λ max Results From Computation where λ min, λ max spec(a). If σ > 1 3, then L(m, s) (K 3 ) 2. Via a computer algorithm in Maple, L(9697, 4), L(30193, 53), L(33121, 2), and L(57401, 7) are Folkman graphs.

16 Works Cited L. Lu, Explicit construction of small Folkman graphs, SIAM J. Discrete Math., 21 (2008), pp