2-Exponents of Two-Coloured Lollipops

Abstract

This paper shows that for an asymmetric primitive two-coloured $(n,s)$-lollipop on $n$ vertices with $s\le n,$ its 2-exponent is at most $(s^2-1)/2+(s + 1)(n - s)$. The $(n,s)$-lollipops whose 2-exponents achieving the bound is characterised and for any asymmetric primitive two-coloured $(n,s)$-lollipop, a simple algorithm to find its exponent is presented. Keywords: Two-coloured digraphs; primitive; 2-exponents; $(n,s)$-lollipops.