The Clique Numbers and Chromatic Numbers of Certain Paley Graphs

On the Saxl graph of a permutation group

“…Also observe that we can construct primitive groups G with the property that ω(G) is arbitrarily large. For instance, ω(G) = q when G = F q 2 :D q+1 is a primitive affine group as in Example 2.4 (see [6]). In view of Lemma 3.6(iii), further examples can be constructed from any family of base-two groups G such that Q(G, 2) → 0 as |G| tends to infinity (see Section 3.4).…”

Pseudo‐random properties of self‐complementary symmetric graphs

“…There are several arguments showing it is ffiffi ffi q p ; see, for example, [5,Theorem 14,Chapter XIII]. If q is a square, it can be shown that !ðPðqÞÞ ¼ ðPðqÞÞ ¼ ffiffi ffi q p , with the subfield of PðqÞ of order ffiffi ffi q p being a maximum clique: a proof is in [6]. This is far larger than the clique number of Gðn; 1=2Þ, which is almost surely approximately 2 log 2 ðnÞ.…”

“…Proof. As in [6], we look for a large subfield which is a complete subgraph. Let F ¼ F q be the underlying field, and F* ¼ Fnf0g be its multiplicative group.…”

Involution Graphs Where the Product of Two Adjacent Vertices Has Order Three

“…If (G, X, S) is connected, then X is both a G-conjugacy class and an X -conjugacy class. [7] Involution graphs where the product of two adjacent vertices has order three 311 PROOF. By Lemma 2.12, (G, X, S) = ( X , X, S).…”

“…Moreover, the two conjugacy classes of S 3 -subgroups of P remain separate conjugacy classes in G. Thus the restriction of to the involutions of P is the Paley graph of GF(9 e ). By [7] the largest clique in the Paley graph of GF(9 e ) has size 3 e . Moreover, [5] proved that such cliques are affine images of subfields GF(3 e ).…”