Recently I am trying to use Talagrand concentration inequality to do something on graphs. I find a version from the book of Molloy and Reed ''Graph Colouring and Probabilistics Method''. I attached a picture from that book below. I am very confused with that whether the bound $E[X]$ for $t$ is necessary. Can we ignore it when apply Talagrand inequality? If we can do so, is there any publication talking about this?