On island sequences of labelings with a condition at distance two

Abstract

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.

Path covering number and L(2,1)-labeling number of graphs

“…The island sequence is the ordered sequence of island cardinalities in nondecreasing order. Figure 1 [2] presents two different λ-labelings of the complete bipartite graph K 2,3 with λ = 5 and ρ = 1, and inducing the same island sequence (2,3). Figure 2 presents two different λ-labelings of the non-connected graph K 5 ∪ K 2 with λ = 8 and ρ = 2, and inducing two different island sequences (1,1,5) and (1,3,3), respectively.…”

“…A graph G is 2-sparse if G contains no pair of adjacent vertices of degree greater than 2. The above notation is first introduced by Adams et al [2] and they solved the above question by studying complements of 2-sparse trees.…”

“…Theorem 4 (cf. Theorem 2.8 in [2])Let T be a 2-sparse tree. If T is neither a path nor a generalized star, then its complement T c is connected and admits at least two different island sequences.…”

“…Theorem 6 (cf. Theorem 3.2 in [2])Let G be a connected 2-sparse graph with m ≥ 1 edges, n vertices, and ℓ leaves. If G is not a cycle, then P (G) = ℓ + m − n.…”

“…Also, when combined with the prior results in Theorem 1 and Theorem 2, it implies that the L(2, 1)-labeling number and hole index for complements of non-path 2-sparse trees and for complements of certain non-cycle 2-sparse graphs are determined. As pointed in [2], further study in two directions is encouraged:…”