Deficiency of forests

Abstract

An edge-magic total labeling of an .n; m/-graph G D .V; E/ is a one to one map from V .G/ [ E.G/ onto the integers f1; 2; : : : ; n C mg with the property that there exists an integer constant c such that .x/ C .y/ C .xy/ D c for any xy 2 E.G/. It is called super edge-magic total labeling if .V .G// D f1; 2; : : : ; ng. Furthermore, if G has no super edge-magic total labeling, then the minimum number of vertices added to G to have a super edgemagic total labeling, called super edge-magic deficiency of a graph G, is denoted by s .G/ [4]. If such vertices do not exist, then deficiency of G will be C1. In this paper we study the super edge-magic total labeling and deficiency of forests comprising of combs, 2-sided generalized combs and bistar. The evidence provided by these facts supports the conjecture proposed by .

Embedding of Supplementary Results in Strong EMT Valuations and Strength

“…Lee and Shah [6] proved this conjecture for trees with upto 17 vertices by the help of a computer. Javed et al [7] provided the SEMT labeling for the 2-sided generalized comb. In [4] Enomoto et al showed that all caterpillars possess SEMT labeling.…”

“…Two-sided generalized comb denoted by Cb ν,υ is de ned in [7] and have proved that it admits SEMT labeling.…”