Every finite group has a normal bi-Cayley graph

Abstract

A graph $\G$ with a group $H$ of automorphisms acting semiregularly on the vertices with two orbits is called a {\em bi-Cayley graph} over $H$. When $H$ is a normal subgroup of $\Aut(\G)$, we say that $\G$ is {\em normal} with respect to $H$. In this paper, we show that every finite group has a connected normal bi-Cayley graph. This improves Theorem~5 of [M. Arezoomand, B. Taeri, Normality of 2-Cayley digraphs, Discrete Math. 338 (2015) 41--47], and provides a positive answer to the Question of the above paper.