A digraph D with p vertices and q arcs is labeled by assigning a distinct integer value g(v) from {0, 1, ..., q} to each vertex v. The vertex values, in turn, induce a value. If the arc values are all distinct then the labeling is called a graceful labeling of a digraph. In this paper, we prove a general result on graceful digraphs of which Du and Sun's conjecture (J. Beijing Univ. Posts Telecommun, 17: 85-88 1994) is a special case. Further, we provide an upper bound for the number of non isomorphic graceful directed cycles obtained from a graceful labeling of the unicycle − → C n .
A digraph D with p vertices and q arcs is labeled by assigning a distinct integer value g(v) from f0, 1, :::, qg to each vertex v. The vertex values, in turn, induce a value g(u, v) on each arc (u, v) where g(u, v) ¼ (g(v) À g(u)) (mod q þ 1) If the arc values are all distinct then the labeling is called a graceful labeling of digraph. In this survey article, we have collected results that we could find interesting on graceful labeling of digraphs.
In this paper, if prime p ≡ 3 (mod 4) is sufficiently large then we prove an upper bound on the number of occurences of any arbitrary pattern of quadratic residues and nonresidues of length k as k tends to ⌈log 2 p⌉. As an immediate consequence, it proves that, there exist a constant c such that, the least nonresidue for such primes is at most c⌈log 2 p⌉.
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