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CORES OF SYMMETRIC GRAPHS

  • PETER J. CAMERON (a1) and PRISCILA A. KAZANIDIS (a2)

Abstract

The core of a graph Γ is the smallest graph Δ that is homomorphically equivalent to Γ (that is, there exist homomorphisms in both directions). The core of Γ is unique up to isomorphism and is an induced subgraph of Γ. We give a construction in some sense dual to the core. The hull of a graph Γ is a graph containing Γ as a spanning subgraph, admitting all the endomorphisms of Γ, and having as core a complete graph of the same order as the core of Γ. This construction is related to the notion of a synchronizing permutation group, which arises in semigroup theory; we provide some more insight by characterizing these permutation groups in terms of graphs. It is known that the core of a vertex-transitive graph is vertex-transitive. In some cases we can make stronger statements: for example, if Γ is a non-edge-transitive graph, we show that either the core of Γ is complete, or Γ is its own core. Rank-three graphs are non-edge-transitive. We examine some families of these to decide which of the two alternatives for the core actually holds. We will see that this question is very difficult, being equivalent in some cases to unsolved questions in finite geometry (for example, about spreads, ovoids and partitions into ovoids in polar spaces).

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Corresponding author

For correspondence; e-mail: P.J.Cameron@qmul.ac.uk

References

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[1]Araújo, João, ‘A group theoretical approach to synchronizing automata and the Černý problem’, Preprint, Lisbon, January 2008.
[2]Arnold, Fredrick and Steinberg, Benjamin, ‘Synchronizing groups and automata’, Theoret. Comput. Sci. 359 (2006), 101110.
[3]Baker, R. D., ‘Partitioning the planes of AG2m(2) into 2-designs’, Discrete Math. 15 (1976), 205211.
[4]Beutelspacher, A., ‘On parallelisms in finite projective spaces’, Geom. Dedicata 3 (1974), 3540.
[5]Cameron, Peter J., Projective and Polar Spaces, QMW Maths Notes, 13 (Queen Mary and Westfield College, London, 1991).
[6]Cameron, Peter J., Permutation Groups, London Mathematical Society Student Texts, 45 (Cambridge University Press, Cambridge, 1999).
[7]Denniston, R. H. F., ‘Some packings of projective spaces’, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 52 (1972), 3640.
[8]Dixon, J. D. and Mortimer, B., Permutation Groups, Graduate Texts in Mathematics, 163 (Springer, New York, 1996).
[9] The Group, ‘ — Groups, Algorithms, and Programming’, Version 4.4.9, 2006; see http://www.gap-system.org.
[10]Godsil, Chris  and Royle, Gordon F., ‘Cores of geometric graphs’, Preprint, Waterloo, ON, Canada, March 2008.
[11]Hahn, Geňa and Tardif, Claude, ‘Graph homomorphisms: structure and symmetry’, in: Graph Symmetry, NATO Advanced Science Institute, Series C, Mathematical and Physical Sciences, 497 (eds. G. Hahn and G. Sabidussi) (Kluwer Academic, Dordrecht, 1997), pp. 107166.
[12]Hell, Pavol and Nešetřil, Jaroslav, Graphs and Homomorphisms (Oxford University Press, Oxford, 2004).
[13]Higman, D. G., ‘Finite permutation groups of rank 3’, Math. Z. 86 (1964), 145156.
[14]Kantor, W. M. and Liebler, R. A., ‘The rank 3 permutation representations of the finite classical groups’, Trans. Amer. Math. Soc. 271 (1982), 171.
[15]Liebeck, M. W., ‘The affine permutation groups of rank three’, Proc. London Math. Soc. (3) 54 (1987), 477516.
[16]Liebeck, M. W. and Saxl, J., ‘The finite primitive permutation groups of rank three’, Bull. London Math. Soc. 18 (1986), 165172.
[17]Lovász, L., ‘Kneser’s conjecture, chromatic number, and homotopy’, J. Combin. Theory Ser. A 25 (1978), 319324.
[18]Neumann, Peter M., ‘Primitive permutation groups and their section-regular partitions’, Michigan Math. J. to appear.
[19]Soicher, L. H., ‘The GRAPE package for GAP’, Version 4.3, 2006,http://www.maths.qmul.ac.uk/∼leonard/grape/.
[20]Thas, J. A., ‘Projective geometry over a finite field’, in: Handbook of Incidence Geometry (ed. F. Buekenhout) (Elsevier, Amsterdam, 1995), pp. 295347.
[21]Welzl, E., ‘Symmetric graphs and interpretations’, J. Combin. Theory B 37 (1984), 235244.
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