For a given graph, we define a digit-sum vertex-labeling as a vertex-labeling with positive integers such that each vertex-label is the sum of digits on all the adjacent vertex-labels. We also define indispensable digits in a nonnegative integer n such that the digit sum of  is 9 times the number of indispensable digits in n. Then, we show there exists a digit-sum vertex-labeling on a graph, if the degree of each vertex on the graph has one indispensable digit; or if the graph is n‑regular and  has k indispensable digits for some positive integers k.

digit sum, casting out nines, vertex-labeling, indispensable digits.