r -regular families of graph automorphisms Robert Jajcay Comenius - PowerPoint PPT Presentation

r -regular families of graph automorphisms Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk .............................. Joint work with Gareth Jones October 11, 2016 Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk .............................. r -regular families of graph automorphisms Joint work

Vertex-Transitive Graphs Definition A graph Γ = ( V , E ) is said to be vertex-transitive if the group Aut (Γ) of automorphisms of Γ acts transitively on V , i.e., u , v ∈ V ∃ ϕ ∈ Aut (Γ) : ϕ ( u ) = v Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk .............................. r -regular families of graph automorphisms Joint work

Cayley Graphs Definition A graph Γ = ( V , E ) is said to be Cayley if the group Aut (Γ) of automorphisms of Γ contains a subgroup that acts regularly on V , i.e., u , v ∈ V ∃ unique ϕ ∈ Aut (Γ) : ϕ ( u ) = v Conjecture (?): # of Cayley graphs of order ≤ n lim > 0 # of v-t graphs of order ≤ n n →∞ Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk .............................. r -regular families of graph automorphisms Joint work

Definition (Sabidussi, Godsil) The Cayley deficiency of a vertex-transitive graph Γ, d (Γ), is the order of the vertex-stabilizer of a smallest vertex-transitive automorphism group of Γ. ◮ Cayley graphs are exactly the vertex-transitive graphs with d (Γ) = 1; ◮ the Petersen graph P of order 10 admits a vertex-transitive group of automorphism of order 20 and none smaller, hence d ( P ) = 2; ◮ there exist vertex-transitive graphs that are arbitrarily far from being Cayley: Kneser and Johnson graphs [Jon&RJ] Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk .............................. r -regular families of graph automorphisms Joint work

Quasi-Cayley Graphs Definition (Gauyacq) A vertex-transitive graph Γ = ( V , E ) is quasi-Cayley iff there exists a regular family of graph automorphisms F satisfying the property that for each pair of vertices u , v ∈ V there exists a unique automorphism ϕ ∈ F mapping u to v , ϕ ( u ) = v . ◮ every quasi-Cayley graph is vertex-transitive ◮ every Cayley graph is quasi-Cayley ◮ Petersen graph is not Cayley but it is quasi-Cayley Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk .............................. r -regular families of graph automorphisms Joint work

Construction of quasi-Cayley graphs ◮ take a Latin square L ; ◮ let the rows of L define permutations ϕ i , 1 ≤ i ≤ n , of { 1 , 2 , . . . , n } : ϕ i ( j ) = L i , j ◮ take G = � ϕ 1 , ϕ 2 , . . . , ϕ n � � n ◮ take E to be a union of the orbits of G in the action on � 2 ◮ the graph ( { 1 , 2 , . . . , n } , E ) is quasi-Cayley but it can also be Cayley Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk .............................. r -regular families of graph automorphisms Joint work

r -regular families Definition (RJ,Jones) A subset F ⊆ S X of permutations of a set of elements X is said to form an r -regular family of permutations on X if for every pair of elements x , y ∈ X there exist exactly r permutations ϕ ∈ F mapping x to y , ϕ ( x ) = y . Definition (RJ, Jones) The quasi-Cayley deficiency of a vertex-transitive graph Γ, r (Γ), is the smallest r for which there exists an r -regular family of automorphisms of Γ. Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk .............................. r -regular families of graph automorphisms Joint work

Quasi-Cayley Deficiency Observation: If Γ = ( V , E ) is a vertex-transitive graph, then Γ admits an r -regular family of automorphisms where r is smaller or equal to the order of the vertex-stabilizer of a smallest vertex-transitive automorphism group of Γ. r (Γ) ≤ d (Γ) r ( P ) = 1 , d ( P ) = 2 Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk .............................. r -regular families of graph automorphisms Joint work

Vertex-transitive graphs which are not quasi-Cayley graphs 1. the Johnson graph J = J ( n , k ) has as its vertex set the set � N � V = of k -element subsets of an n -element set N , with k vertices v and w adjacent if and only if | v ∩ w | = k − 1 2. the Kneser graph K = K ( n , k ) has the same vertex-set as J ( n , k ) with the adjacency defined between any two k -element subsets that are disjoint 3. the distance i graph J ( n , k ) i has the same vertex set and with adjacency defined by | v ∩ w | = k − i 4. the merged Johnson graph J ( n , k ) I = ∪ i ∈ I J ( n , k ) i for some non-empty proper subset I of { 2 , . . . , k } Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk .............................. r -regular families of graph automorphisms Joint work

Vertex-transitive graphs not admitting r -regular families for small r ’s Lemma (RJ,Jones) Let M be an r · C ( n , k ) × n matrix with entries from { 1 , 2 , . . . , n } satisfying properties: 1. each row contains all of { 1 , 2 , . . . , n } , or, in other words, each i ∈ { 1 , 2 , . . . , n } appears exactly once in each row; 2. the matrix formed by any subset of k columns of M contains each k-subset of { 1 , 2 , . . . , n } (as a row) exactly r times. Then � n − 1 � 1. k | r k − 1 � n − 2 � 2. k | 2 r k − 2 Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk .............................. r -regular families of graph automorphisms Joint work

Vertex-transitive graphs not admitting r -regular families for small r ’s Corollary (RJ,Jones) Let p be a prime, k = p and n = ℓ p for a positive integer ℓ > 2 . Then J ( n , k ) does not admit any r-regular family for 1 ≤ r < k. Corollary (RJ,Jones) For every k ≥ 1 , there exist infinitely many vertex-transitive graphs Γ for which the smallest r (Γ) > k. Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk .............................. r -regular families of graph automorphisms Joint work

Classification of merged Johnson graphs that are Cayley Theorem (Jones, RJ) Let 2 ≤ k ≤ n / 2 and let I be a non-empty subset of { 1 , . . . , k } . Then the merged Johnson graph J = J ( n , k ) I is a Cayley graph of a group G if and only if one of the following holds: 1. n is a prime power, n ≡ 3 mod (4) and k = 2 , with any I, and G ∼ = AHL 1 ( F ) acting on some Dickson near-field N = F of order n; 2. n = 8 and k = 3 , with any I, and G ∼ = AGL 1 (8) acting on the finite field N = F 8 ; 3. n = 32 and k = 3 , with any I, and G ∼ = A Γ L 1 (32) acting on the finite field N = F 32 ; 4. I = { 1 , . . . , k } , with any n and k, and G any group of order � n � , acting on itself by right multiplication; k 5. k = n / 2 and I = { k } or { 1 , . . . , k − 1 } , with G any group of � n � order , acting on itself by right multiplication. k Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk .............................. r -regular families of graph automorphisms Joint work

Theorem (Jones,RJ) Let 2 ≤ k ≤ n / 2 and let I be a non-empty subset of { 1 , . . . , k } . Then G is a 2 -regular group of automorphisms of the graph J = J ( n , k ) I if and only if one of the following holds: 1. n is a prime power, k = 2 , with any I, and G ∼ = AGL 1 ( F ) for some near-field N = F of order n; 2. n = 6 and k = 3 , where G ∼ = AGL 1 (5) × S 2 , with any I, and AGL 1 (5) acting naturally on the projective line N = P 1 ( F 5 ) , fixing ∞ , and S 2 generated by complementation in N; 3. n = 10 , k = 5 and I = { 1 , 4 } , { 2 , 3 } , { 1 , 4 , 5 } or { 2 , 3 , 5 } , where G ∼ = PSL 2 (8) 4. n is even, k = n / 2 and I = { k } or { 1 , . . . , k − 1 } , where G is � n � any group of order 2 with a non-normal subgroup H of k order 2 , acting on the cosets of H; � n � 5. I = { 1 , . . . , k } , where G is any group of order 2 with a k non-normal subgroup H of order 2 , acting on the cosets of H. Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk .............................. r -regular families of graph automorphisms Joint work

If k ≥ 6, or if 2 ≤ k ≤ 5 and n avoids an easily described subset of N of asymptotic density 0, then the only k -homogeneous groups of degree n , and hence the only vertex-transitive groups of automorphisms of J ( n , k ), are the symmetric group S n and the alternating group A n . d ( J ( n , k )) ≥ n ! 2 for the majority of Johnson graphs. However, we could not find an infinite family of graphs J ( n , k ) for which we could prove that r ( J ( n , k )) is much smaller than d (Γ). Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk .............................. r -regular families of graph automorphisms Joint work

Praeger-Xu graphs Theorem (RJ,Potoˇ cnik,Wilson) 1. Every Praeger-Xu graph is quasi-Cayley, r ( PX ( n , k )) = 1 ; 2. for every positive integer k, there exists a Praeger-Xu graph Γ = PX ( n , k ) with the property d ( PX ( n , k )) > k. Robert Jajcay Comenius University and University of Primorska robert.jajcay@fmph.uniba.sk .............................. r -regular families of graph automorphisms Joint work