Two Conjectures on Graceful Digraphs

Graceful labeling of digraphs—a survey

“…If a degree balanced digraph D is graceful then number of arcs in D is even.By using Friedander et al[8] results Hegde and Shivarajkumar[12] proved the following Theorem 4.16. The digraph Dðn 1 , n 2 , :::, n k ; m 1 , m 2 , :::, m k Þ is graceful if and only if P k i¼1 n i m i is even.…”

“…Dðn 1 , n 2 , :::, n k ; m 1 , m 2 , :::, m k Þ where k is any positive integer denotes the digraph having n 1 unicycles of length m 1 , n 2 unicycles of length m 2 , … , n k unicycles of length m k having one common vertex.By solving the corresponding syetem of simultaneous congruences governed by degree balanced digraphs, Hegde and Shivarajkumar[12] proved the following.…”