On graceful unicyclic wheels

Abstract

A digraph D with e edges is labeled by assigning a distinct integer value ?(?) from (0, l,...,e) to each vertex ?. The vertex values, in turn, induce a value 0(?) ?) on each edge (?, ?) where ? (?, ?) = ?(?) - ? (?)mod(e + 1). If the edge values are all distinct and nonzero, then the labeling is called a graceful labeling of a digraph. In 1985, Bloom and Hsu conjectured that " All unicyclic wheels are gracefulIn this paper we prove the conjecture. Copyright 2014, Charles Babbage Research Centre.