a b s t r a c tAn L(2, 1)-labeling of a graph G is a function f from the vertex set of G to the set of where d(x, y) denotes the distance between the pair of vertices x, y. The lambda number of G, denoted λ(G), is the minimum range of labels used over all L(2,1)-labelings of G. An L(2,1)-labeling of G which achieves the range λ(G) is referred to as a λ-labeling. A hole of an L(2,1)-labeling is an unused integer within the range of integers used. The hole index of G, denoted ρ(G), is the minimum number of holes taken over all its λ-labelings. An island of a given λ-labeling of G with ρ(G) 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 L(2,1)-labelings, SIAM J. Discrete Math. 19 (2005) 208-223] inquired about the existence of a connected graph G with ρ(G) ≥ 1 possessing two λ-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.

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For a positive integer [Formula: see text], a radio k-labeling of a graph [Formula: see text] is a function [Formula: see text] from its vertex set to the non-negative integers such that for all pairs of distinct vertices [Formula: see text] and [Formula: see text], we have [Formula: see text] where [Formula: see text] is the distance between the vertices [Formula: see text] and [Formula: see text] in [Formula: see text]. The minimum span over all radio [Formula: see text]-labelings of [Formula: see text] is called the radio k-chromatic number and denoted by [Formula: see text]. The most extensively studied cases are the classic vertex colorings ([Formula: see text]), [Formula: see text](2,1)-labelings ([Formula: see text]), radio labelings ([Formula: see text], the diameter of [Formula: see text]), and radio antipodal labelings ([Formula: see text]. Determining exact values or tight bounds for [Formula: see text] is often non-trivial even within simple families of graphs. We provide general lower bounds for [Formula: see text] for all cycles [Formula: see text] when [Formula: see text] and show that these bounds are exact values when [Formula: see text].

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