Yes, you can give $15,000 per year to anyone — children, friends, anyone — without reporting it, without it becoming income to them. It’s your money, so feel free to give it away. BUT, if these are “minor” children (under the age of 18 in most states), you are probably better off opening a 529 college savings account and letting the money grow tax-free for college. Search my site for articles/columns on this topic. Also note, if they are minors you can give them cash, but for larger amounts you should open a “custodial account” at a bank — and that can impact their ability to get financial aid for college.
Of course, if by “children” you mean your adult children, you can just write them a check. In any case, the money will be tax-free to them, and a gift from you, but not a tax deduction to you.