On island sequences of labelings with a condition at distance two

Abstract: An -labeling of a graph is a function from the vertex set of to the set of nonnegative integers such that if , and if , where denotes the distance between the pair of vertices . The lambda number of , denoted , is the minimum range of labels used over all (2,1)-labelings of . An (2,1)-labeling of which achieves the range is referred to as a -labeling. A hole of an (2,1)-labeling is an unused integer within the range of integers used. The hole index of , denoted , is the minimum number of holes taken over all its -labelings. An island of a given -labeling of with 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 (2,1)-labelings, SIAM J. Discrete Math. 19 (2005) 208–223] inquired about the existence of a connected graph with possessing two -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. [Copyright &y& Elsevier]