An extremal problem on non-full colorable graphs Original Research Article

For a given graph G of order n, a k-image-labelling is defined as a function image such that image when image and image when image. The image-labelling number of G, denoted by image, is the smallest number k such that G has a k-image-labelling. The hole index image of G is the minimum number of integers not used in a image-image-labelling of G. We say G is full-colorable if image; otherwise, it will be called non-full colorable. In this paper, we consider the graphs with image and image, where m is a positive integer. Our main work generalized a result by Fishburn and Roberts [No-hole image-colorings, Discrete Appl. Math. 130 (2003) 513–519].