On island sequences of labelings with a condition at distance two Original Research Article

An image-labeling of a graph image is a function image from the vertex set of image to the set of nonnegative integers such that image if image, and image if image, where image denotes the distance between the pair of vertices image. The lambda number of image, denoted image, is the minimum range of labels used over all image(2,1)-labelings of image. An image(2,1)-labeling of image which achieves the range image is referred to as a image-labeling. A hole of an image(2,1)-labeling is an unused integer within the range of integers used. The hole index of image, denoted image, is the minimum number of holes taken over all its image-labelings. An island of a given image-labeling of image with image holes is a maximal set of consecutive integers used by the labeling. Georges and Mauro [J.P. Georges, D.W. Mauro, On the structure of graphs with non-surjective image(2,1)-labelings, SIAM J. Discrete Math. 19 (2005) 208–223] inquired about the existence of a connected graph image with image possessing two image-labelings with different ordered sequences of island cardinalities. This paper provides an infinite family of such graphs together with their lambda numbers and hole indices. Key to our discussion is the determination of the path covering number of certain 2-sparse graphs, that is, graphs containing no pair of adjacent vertices of degree greater than 2.