Place the eggs in a medium saucepan with water to cover and bring to a boil. Remove from the heat, cover the pan, and let stand for 20 minutes. Pour off the hot water and refill the saucepan with cold water. Crack the eggshells all over and let them sit in the cold water for 5 minutes. Peel the eggs, cover and chill for at least 1 hour.
Halve the eggs lengthwise. Carefully remove the yolks and transfer them to a small bowl. Mash the yolks with a fork, then stir in the mayonnaise, sweet pickle relish and yellow mustard. Season with salt and pepper. Scoop a spoonful of the mixture into each egg white half. Sprinkle the tops with paprika.
With this method for hard boiling eggs, you won't have any green in your yolk!
Recipe adapted from Georgia Cooking in an Oklahoma Kitchen by Trisha Yearwood (c) Clarkson Potter 2008