Name
Abstract | ||
|---|---|---|
An L(2,1)-labeling of a graph G is a function f from the vertex set of G to the set of nonnegative integers such that |f(x)-f(y)|=2 if d(x,y)=1, and |f(x)-f(y)|=1 if d(x,y)=2, where d(x,y) denotes the distance between the pair of vertices x,y. The lambda number of G, denoted @l(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 @l(G) is referred to as a @l-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 @r(G), is the minimum number of holes taken over all its @l-labelings. An island of a given @l-labeling of G with @r(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 @r(G)=1 possessing two @l-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. |
Year | DOI | Venue |
|---|---|---|
2010 | 10.1016/j.dam.2009.08.005 | Discrete Applied Mathematics |
Keywords | Field | DocType |
1)-labeling,2-sparse graph,w. mauro,path covering number,hole index,island sequence,minimum number,island cardinalities,l (2,lambda number,non-surjective l,minimum range,connected graph,graph g,1)-coloring,labeling,indexation,l,discrete mathematics,graph theory | Integer,Discrete mathematics,Combinatorics,Maximal set,Vertex (geometry),Covering number,Cardinality,Connectivity,Mathematics,Lambda,Integer sequence | Journal |
Volume | Issue | ISSN |
158 | 1 | Discrete Applied Mathematics |
Citations | PageRank | References |
2 | 0.38 | 12 |
Authors | ||
4 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Sarah Spence Adams | 1 | 55 | 7.61 |
| Alex Trazkovich | 2 | 2 | 0.38 |
| Denise Sakai Troxell | 3 | 76 | 9.24 |
| Bradford Westgate | 4 | 9 | 1.28 |