Decompositions into linear forests and difference labelings of graphs

Abstract

Difference labelings of a graph are realized by assigning distinct integer values to each vertex and then associating with each edge uv, the absolute difference of those values assigned to its endvertices u and v. Labelings of this type include bandsize labelings, in which the cardinality of the edge label set is minimized, and graceful labelings, in which the cardinality of the edge label set equals the number of edges. Three new classes of difference labeling problems are defined here, based on the fact that the subsets of edges with common edge labels form linear forests that partition the edge set. In particular, the following three questions are addressed: (1) Given graph G with edge set E( G) and a decomposition D of G into linear forests, does there exist a labeling of the vertices of G for which D is a common-weight decomposition? (2) Given graph G with edge set E( G) and a collection of linear forests F 1, F 2, … , F k containing a total of | E| edges, does there exist a common-weight decomposition of G whose parts are respectively isomorphic to F 1, F 2, … , F k ? (3) Given graph G with edge set E( G) and a set of integers m 1, m 2, … , m k whose sum is | E|, does there exist a common-weight decomposition of G whose parts contain | E i | = m i (1 ⩽ i ⩽ k) edges? Answers to these questions for certain classes of graphs are found, tools for studying these questions are developed, and some open problems are formulated.