Seminars in April 2018

  • Hong Liu, Two conjectures in Ramsey-Turán theory
    Two conjectures in Ramsey-Turán theory
    Hong Liu
    Mathematics Institute, University of Warwick, Warwick, UK
    2018/4/10 Tue 5PM
    Given graphs H1,…, Hk, a graph G is (H1,…, Hk)-free if there is a k-edge-colouring of G with no Hi in colour-i for all i in {1,2,…,k}. Fix a function f(n), the Ramsey-Turán function rt(n,H1,…,Hk,f(n)) is the maximum size of an n-vertex (H1,…, Hk)-free graph with independence number at most f(n). We determine rt(n,K3,Ks,δn) for s in {3,4,5} and sufficiently small δ, confirming a conjecture of Erdős and Sós from 1979. It is known that rt(n,K8,f(n)) has a phase transition at f(n)=Θ(√(n\log n)). We prove that rt(n,K8,o(√(n\log n)))=n2/4+o(n2), answering a question of Balogh, Hu and Simonovits. The proofs utilise, among others, dependent random choice and results from graph packings. Joint work with Jaehoon Kim and Younjin Kim.
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  • Mark Siggers, The reconfiguration problem for graph homomorpisms
    The reconfiguration problem for graph homomorpisms
    Mark Siggers
    Department of Mathematics, Kyungpook National University, Daegu
    2018/04/03 Tue 5PM
    For problems with a discrete set of solutions, a reconfiguration problem defines solutions to be adjacent if they meet some condition of closeness, and then asks for two given solutions it there is a path between them in the set of all solutions.
    The reconfiguration problem for graph homomorphisms has seen fair activity in recent years. Fixing a template, the problem Recol(H) for a graph H asks if one can get from one H-colouring of a graph G to another by changing one vertex at a time, always remaining an H-colouring. If the changed vertex has a loop, it must move to an adjecent vertex
    Depending on H, the problem seems to be either polynomial time solvable or PSPACE-complete. We discuss many recent results in the area and work towards conjecturing for which H the problem is PSPACE-complete.
    This is joint work with Rick Brewster, Jae-baek Lee, Ben Moore and Jon Noel.
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