Charlotte Scott Centre for Algebra

On the 27th of March 2019, **Dr Joanna Fawcett (Imperial College, London) **visited Charlotte Scott Centre for Algebra in Lincoln and gave a talk on “Tree-homogeneous graphs”.

*Abstract*: Let $latex X$ be a class of graphs. A graph $latex Gamma$ is $latex X$-homogeneous if every graph isomorphism $latex varphi:Delta_1to Delta_2$ between induced subgraphs $latex Delta_1$ and $latex Delta_2$ of $latex Gamma$ such that $latex Delta_1in X$ extends to an automorphism of $latex Gamma$. For example, if $latex X={K_1}$, then $latex X$-homogeneity is vertex-transitivity, and if $latex X={K_2}$, then $latex X$-homogeneity is arc-transitivity. A graph is *tree-homogeneous* if it is $latex X$-homogeneous where $latex X$ is the class of trees. We discuss some recent progress on classifying the finite tree-homogeneous graphs, as well as some connections with certain highly symmetric incidence geometries called partial linear spaces.

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