One modulo N gracefulness of acyclic graphs

AUTHOR AND
AFFILIATION

V. RAMACHANDRAN1 and C. SEKAR2

KEYWORDS:

Graceful, modulo N graceful, Path, Sm,n , Sm,n Sm,n, Caterpillar, Star, Lobster, Banana tree and Rooted tree ofheight two AMS Subject Classification (2010):05C78

Issue Date:

December 2013

Pages:

ISSN:

2319-8044 (Online) – 2231-346X (Print)

Source:

Vol.25 – No.3

PDF

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DOI:

jusps-A

ABSTRACT:

A function f is called a graceful labelling of a graph G with q edges if f is an injection from the vertices of G to the set (0, 1, 2,…, q) such that, when each edge xy is assigned the label | f(x) — f(y)|, the resulting edge labels are distinct. A graph G is said to be one modulo N graceful (where N is a positive integer) if there is a function  from the vertex set
of G to {0, 1. N, (N + 1), 2N, (2N + 1),…, N(q – 1). N(q -1) + 1} in such a way that (i)  is 1-1 (ii)  induces a bijection * from the edge set of G to {1, N+1, 2N +1,…, N(q – 1)+1} where *(uv) = |(u) - (v)|. In this paper we prove that the acyclic graphs viz. Paths, Caterpillars, Stars and S2,n S2,n are one modulo N graceful for all positive integer N; Lobsters, Banana trees and Rooted tree of height two are one modulo N graceful for N > 1. where Sm,n Sm,n is a graph obtained by identifying one pendant vertex of each Sm,n. This is a fire cracker of subdivisioned stars.

Copy the following to cite this Article:

V. RAMACHANDRAN1 and C. SEKAR2, “One modulo N gracefulness of acyclic graphs”, Journal of Ultra Scientist of Physical Sciences, Volume 25, Issue 3, Page Number , 2016


Copy the following to cite this URL:

V. RAMACHANDRAN1 and C. SEKAR2, “One modulo N gracefulness of acyclic graphs”, Journal of Ultra Scientist of Physical Sciences, Volume 25, Issue 3, Page Number , 2016

Available from: http://www.ultrascientist.org/paper/189/


A function f is called a graceful labelling of a graph G with q edges if f is an injection from the vertices of G to the set (0, 1, 2,…, q) such that, when each edge xy is assigned the label | f(x) — f(y)|, the resulting edge labels are distinct. A graph G is said to be one modulo N graceful (where N is a positive integer) if there is a function  from the vertex set
of G to {0, 1. N, (N + 1), 2N, (2N + 1),…, N(q – 1). N(q -1) + 1} in such a way that (i)  is 1-1 (ii)  induces a bijection * from the edge set of G to {1, N+1, 2N +1,…, N(q – 1)+1} where *(uv) = |(u) - (v)|. In this paper we prove that the acyclic graphs viz. Paths, Caterpillars, Stars and S2,n S2,n are one modulo N graceful for all positive integer N; Lobsters, Banana trees and Rooted tree of height two are one modulo N graceful for N > 1. where Sm,n Sm,n is a graph obtained by identifying one pendant vertex of each Sm,n. This is a fire cracker of subdivisioned stars.