Maximum gap labelings of graphs

• Published in: Information processing letters Vol. 111; no. 4; pp. 169 - 173
• Main Authors: Feder, Tomás, Subi, Carlos
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 15.01.2011; Elsevier; Elsevier Sequoia S.A
• Subjects: Algorithmics. Computability. Computer arithmetics; Algorithms; Applied sciences; Approximation; Approximation algorithms; Approximations and expansions; Computer science; control theory; systems; Exact sciences and technology; Gaps; Graceful labelling of trees; Graph algorithms; Graph theory; Graphs; Information retrieval. Graph; Integers; Joints; Labeling; Marking; Mathematical analysis; Mathematics; Maximum gap in graph; Minimum edge deletion bipartition; Miscellaneous; Sciences and techniques of general use; Studies; Theoretical computing; Trees; Approximation algorithms; Graceful labelling of trees; Minimum edge deletion bipartition; Maximum gap in graph; Edge(graph); Computer theory; Polynomial approximation; Label; Approximation algorithm; Polynomial time; Assignment; Labelled graph; Integer; Gap problem; Input; Maximum; Information processing; Deletion; NP complete problem; Bipartite graph; Graph labelling; Algorithm analysis; Graph algorithm; Deterministic algorithms
• ISSN: 0020-0190; 1872-6119
• DOI: 10.1016/j.ipl.2010.11.010

Given a graph, we define a base set to be a set of integers of size equal to the number of vertices in the graph. Given a graph and a base set, a labeling of the graph from the base set is an assignment of distinct integers from the base set to the vertices of the graph. The gap of an edge in a labeled graph is the absolute value of the difference between the labels of its endpoints. The gap of a labeled graph is the sum of the gaps of its edges. The maximum gap graph labeling problem takes as input a graph and a base set and maximizes the gap of the graph over all possible labelings from the base set. We show that this problem is NP-complete even when the base set is restricted to consecutive integers. We also show that this restricted case has polynomial time approximations that achieve a factor of 2/3 for trees, of 1/2 for bipartite graphs, and of 1/4 for general graphs, with a deterministic algorithm, while an expected factor of 1/3 for general graphs is achieved with a randomized algorithm. The case of general base sets is approximated within an expected factor of 1/16 for general graphs with a randomized polynomial time algorithm. We finally give a polynomial time algorithm that solves the maximum gap graph labeling problem for a graph that has bounded degree and bounded treewidth. The maximum graph labeling problem shows connections with the graceful tree conjecture. ► We assign labels to vertices to maximize edge difference sum. ► The label set is fixed ahead of time. ► We give polynomial time exact and approximation algorithms. ► We give NP-completeness proofs.