### On Vertex-transitive Cayley Graphs of Finite Transformation Semigroups with Restricted Range

#### Abstract

Let \(T(X)\) be the semigroup of all transformations on a set \(X\). For a non-empty subset \(Y\) of \(X\), denoted by \(T(X,Y)\) the subsemigroup of \(T(X)\) consisting of all transformations whose range is contained in \(Y\). Kelarev and Praeger in [9] gave necessary and sufficient conditions for all vertex-transitive Cayley graphs of semigroups. In this paper, we give similar descriptions for all vertex-transitive Cayley graphs of \(T(X,Y)\).

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