### Distance labelings: a generalization of Langford sequences

#### Abstract

A Langford sequence of order *m* and defect *d* can be identified with a labeling of the vertices of a path of order 2*m* in which each label from *d* up to *d* + *m* − 1 appears twice and in which the vertices that have been labeled with *k* are at distance *k*. In this paper, we introduce two generalizations of this labeling that are related to distances. The basic idea is to assign nonnegative integers to vertices in such a way that if *n* vertices (*n* > 1) have been labeled with *k* then they are mutually at distance *k*. We study these labelings for some well known families of graphs. We also study the existence of these labelings in general. Finally, given a sequence or a set of nonnegative integers, we study the existence of graphs that can be labeled according to this sequence or set.

#### Keywords

ISSN: 1855-3974

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