Cores of Vertex-Transitive Graphs Ricky Rotheram Submitted in total fulfilment of the requirements of the degree of Master of Philosophy October 2013 Department of Mathematics and Statistics The University of Melbourne Parkville, VIC 3010, Australia Produced on archival quality paper Abstract The core of a graph Γ is the smallest graph Γ∗ for which there exist graph homomor- phisms Γ ! Γ∗ and Γ∗ ! Γ. Thus cores are fundamental to our understanding of general graph homomorphisms. It is known that for a vertex-transitive graph Γ, Γ∗ is vertex-transitive, and that jV (Γ∗)j divides jV (Γ)j. The purpose of this thesis is to determine the cores of various families of vertex-transitive and symmetric graphs. We focus primarily on finding the cores of imprimitive symmetric graphs of order pq, where p < q are primes. We choose to investigate these graphs because their cores must be symmetric graphs with jV (Γ∗)j = p or q. These graphs have been completely classified, and are split into three broad families, namely the circulants, the incidence graphs and the Maruˇsiˇc-Scapellato graphs. We use this classification to determine the cores of all imprimitive symmetric graphs of order pq, using differ- ent approaches for the circulants, the incidence graphs and the Maruˇsiˇc-Scapellato graphs. Circulant graphs are examples of Cayley graphs of abelian groups. Thus, we generalise the approach used to determine the cores of the symmetric circulants of order pq, and apply it to other Cayley graphs of abelian groups. Doing this, we show that if Γ is a Cayley graph of an abelian group, then Aut(Γ∗) contains a transitive subgroup generated by semiregular automorphisms, and either Γ∗ is an odd cycle or girth(Γ∗) ≤ 4. Consequently, we show that Γ∗ is not a cubic symmetric graph ∗ ∼ ∗ (unless Γ = K4) or 3-arc-transitive (unless Γ is an odd cycle). We also provide a partial confirmation of a conjecture by Samal, on the cores of Cayley graphs of elementary abelian groups. Declaration This is to certify that: (i) the thesis comprises only my original work towards the MPhil; (ii) due acknowledgement has been made in the text to all other material used; and (iii) the thesis is less than 50 000 words in length. Ricky Rotheram Acknowledgements My deepest and sincerest gratitude goes to my primary supervisor, Associate Pro- fessor Sanming Zhou, for his guidance and supervision throughout my work on this thesis. I am especially grateful to Sanming for introducing me to vertex-transitive and symmetric graphs, as well as graph homomorphisms and the cores of graphs. I also thank Sanming for being supportive during some difficult periods, both in my academic and personal lives. I also wish to thank my co-supervisor, Professor Gordon Royle, for being a courteous host during my visit to the University of Western Australia in Semester 2 of 2010. I am especially grateful to Gordon for suggesting that I look at primitive Cayley graphs of elementary abelian groups. This suggestion led directly to Section 7.2, and specifically to Theorem 7.2.7. I acknowledge the support of an Australian Postgraduate Award from the Uni- versity of Melbourne. Finally, I wish to thank my parents, who have provided me with constant en- couragement and support over the course of my education. Without their support, this thesis could not have been finished. List of Key Symbols Ω; Ξ; ∆ Sets ∆ ⊆ Ω ∆ is a subset of Ω ∆ × Ξ Cartesian product of ∆ and Ξ ∆l Cartesian product of l copies of ∆ G; H; K Groups H ≤ GH is a subgroup of G H E GH is a normal subgroup of G hSi Subgroup of G generated by subset S 1G Identity element of G K × H Direct product of H by K Kl Direct product of l copies of K K o H Semidirect product of K by H K wr ΞH Wreath product of K by H, with H acting on Ξ G(α) G-orbit containing α G((α; β)) G-orbital containing (α; β) Gα Stabiliser of α in G i ii G∆ Setwise stabiliser of ∆ in G x(∆) Image of ∆ under x 2 G Sym(Ω) Symmetric group of permutations of Ω Alt(n) Alternating group Zn Additive group of integers mod n ∗ Zn Multiplicative group of integers mod n ∗ H(p; r) Unique subgroup of Zp of order r l Zp Elementary abelian group GF (n) Finite field with n elements GF(n)∗ Multiplicative group of GF(n) M10; M11; M22; M23 Mathieu groups SL(n; q) Special linear group Sp(n; q) Symplectic group GL(n; q) General linear group AGL(n; q) Affine general linear group ΓL(n; q) Semilinear group ΓSp(n; q) Symplectic semilinear group PSL(n; q) Projective special linear group PSU(n; q) Projective special unitary group PSp(n; q) Projective symplectic group PΓSp(n; q) Projective symplectic semilinear group H(11) Unique 2-(11; 5; 2) design iii PG(d − 1; n)( d − 1)-dimensional projective space over GF (n) V (n; q) n-dimensional symplectic vector space over GF (q) Wn−1(q) Classical polar space with nondegenerate, alternating bilinear form on V (n; q) Γ; Ψ; Λ Graphs V (Γ) Vertex set of Γ E(Γ) Edge set of Γ A(Γ) Arc set of Γ NΓ(u) Neighbourhood of u in Γ Aut(Γ) Full automorphism group of Γ φ, Graph homomorphisms φ(Γ) Homomorphic image of Γ under φ φ−1(u) Fibre of φ P Partition of V (Γ) Γ=P Quotient graph of Γ with respect to P Ψ $ Λ Ψ and Λ are homomorphically equivalent φ jΛ Restriction of φ : V (Γ) ! V (Λ) to the subgraph Λ of Γ Γ∗ Core of Γ Kn Complete graph on n vertices Cn Cycle of length n Pn+1 Path of length n Γ Complement of Γ iv Kn Empty graph with n vertices s K(r; s) Kneser graph for integers r; s, with 1 ≤ r < 2 Ψ s Circular graph for integers r; s with 0 < r ≤ s r val(Γ) Valency of the regular graph Γ α(Γ) Independence number of Γ I(Γ) Set of all independent sets of Γ I(Γ; x) Set of all independent sets of Γ containing vertex x !(Γ) Clique number of Γ δ(Γ; t) Maximum number of vertices in an induced subgraph of Γ with no complete subgraph of order t dΓ(u; v) Distance between u and v in Γ diam(Γ) Diameter of Γ girth(Γ) Girth of Γ oddg(Γ) Odd-girth of Γ χ(Γ) Chromatic number of Γ χC (Γ) Circular chromatic number of Γ χf (Γ) Fractional chromatic number of Γ Γ2Ψ Cartesian product of Γ and Ψ Γ × Ψ Categorical product of Γ and Ψ Γ Ψ Strong product of Γ and Ψ Γ [Ψ] Lexicographic product of Γ and Ψ Ψ [Λ] − dΨ Deleted lexicographic product of Ψ and Λ v n 2i=1Γi Cartesian product of n graphs n ×i=1Γi Categorical product of n graphs n i=1Γi Strong product of n graphs Γn n-th power of Γ with respect to the categorical product Cay(G; S) Cayley graph of group G, S ⊂ G G(p; r) Symmetric circulant of order p (where p is prime), valency r G(2q; r) Bipartite, symmetric circulant G(pq; ; r; s; u) General symmetric circulant of order pq D t − (v; k; λ) design P Set of points of design D B Set of blocks of design D X(D) Incidence graph of design D X0(D) Complementary incidence graph of design D Γ(a; m; S; U) Maruˇsiˇc-Scapellato graph of order m(2a + 1) Σ System of imprimitivity of V (Γ), with Γ ∼= Γ(a; m; S; U), under SL(2; 2a) ≤ Aut(Γ) a Bx Block of Σ containing all vertices (x; r), with x 2 PG(1; 2 ) Fs Fermat number List of Tables 4.1 Symmetric Circulant Graphs of order pq, p < q, and their Cores . 43 4.2 Incidence and Maruˇsiˇc-Scapellato Graphs of order pq, p < q, and their Cores . 44 vi List of Figures 2.1 The categorical product C5 × K3..................... 15 2.2 The lexicographic product C5 [K3]. ................... 16 2.3 The deleted lexicographic product C5 [K3] − 3C5............ 18 4.1 The symmetric prime order circulant G(13; 4). 45 4.2 The symmetric circulant G(21; 3; 2; 6), with t = 3, a = 2 and c = 2. 48 4.3 The Maruˇsiˇc-Scapellato graph Γ(2; 3; ;; f0g). 52 vii Contents List of Key Symbols . .i List of Tables . vi List of Figures . vii 1 Introduction 1 1.1 Introduction . .1 1.2 Problems to be studied . .2 1.3 Main results and structure of the thesis . .3 2 Notation, definitions and preliminaries 6 2.1 Permutation groups, partitions, blocks and primitivity . .6 2.2 Graphs . 11 2.3 Graph homomorphisms . 19 2.3.1 General graph homomorphisms . 19 2.3.2 Retractions, cores and homomorphic equivalence . 21 2.3.3 Homomorphisms of vertex-transitive graphs . 23 2.3.4 Cores of graph products . 26 3 Literature review 28 3.1 Graph homomorphisms . 29 3.1.1 Existence and nonexistence of graph homomorphisms . 29 3.1.2 Circular colourings . 30 3.1.3 Fractional colourings . 32 3.1.4 Graph products . 34 3.2 Retracts and cores . 37 3.2.1 General retracts and cores . 37 viii ix 3.2.2 Retracts and cores of vertex-transitive graphs . 38 3.2.3 Retracts and cores of graph products . 39 4 Cores of imprimitive, symmetric pq graphs 41 4.1 Imprimitive symmetric graphs of order pq, and their cores .