In this paper we study a technique to transform α-labeled trees into ρ-labeled forests. We use this result to prove that the complete graph K2n+1 can be decomposed into these types of forests. In addition we show a robust family of trees that admit ρ-labelings, we use this result to describe the set of all trees for which a ρ-labeling must be found to completely solve Kotzig’s conjecture about ...

A graceful n-permutation is a graceful labeling of an n-vertex path Pn. In this paper we improve the asymptotic lower bound on the number of such permutations from Ω((5/3)n) to Ω(2.37n). This is a computer-assisted proof based on an effective algorithm that enumerates graceful n-permutations. Our algorithm is also presented in detail. 1. Graceful graphs and permutations Let G = (V,E) be an undi...

We prove that if G is a 2r-regular edge graceful (p, q) graph with (r, kp) = 1 then kG is edge graceful for odd k. We also prove that for certain specific classes of 2r-regular edge graceful graphs it is possible to drop the requirement that (r, kp) = 1

A Smarandache-Fibonacci triple is a sequence S(n), n ≥ 0 such that S(n) = S(n − 1) + S(n − 2), where S(n) is the Smarandache function for integers n ≥ 0. Clearly, it is a generalization of Fibonacci sequence and Lucas sequence. Let G be a (p, q)-graph and {S(n)|n ≥ 0} a Smarandache-Fibonacci triple. An bijection f : V (G) → {S(0), S(1), S(2), . . . , S(q)} is said to be a super Smarandache-Fibo...

In this work some new odd graceful graphs are investigated. We prove that the graph obtained by fusing all the n vertices of Cn of even order with the apex vertices of n copies of K1,m admits odd graceful labeling. In addition to this we show that the shadow graphs of path Pn and star K1,n are odd graceful graphs.

Graph labeling is considered as one of the most interesting areas in graph theory. A for a simple G (numbering or valuation), an association non -negative integers to vertices G  (vertex labeling) edges (edge both them. In this paper we study graceful k- uniform hypertree and define condition corresponding tree be graceful. if minimum difference vertices’ labels each edge dis...

The problem of finding a graceful labelling of a graph, or proving that the graph is not graceful, has previously been modelled as a CSP. A new and much faster CSP model of the problem is presented, with several new results for graphs whose gracefulness was previously unknown. Several classes of graph that are conjectured to be graceful only for small instances are investigated: after a certain...