Interval edge-colorings of K1,m,nK1,m,n

An edge-coloring of a graph GG with colors 1,…,t1,…,t is an interval tt-coloring if all colors are used, and the colors of edges incident to each vertex of GG are distinct and form an interval of integers. A graph GG is interval colorable if it has an interval tt-coloring for some positive integer tt. In this note we prove that K1,m,nK1,m,n is interval colorable if and only if gcd(m+1,n+1)=1gcd(m+1,n+1)=1, where gcd(m+1,n+1)gcd(m+1,n+1) is the greatest common divisor of m+1m+1 and n+1n+1. It settles in the affirmative, a conjecture of Petrosyan.