On the beta-number of forests with isomorphic components

Abstract

The beta-number, β (G), of a graph G is defined to be either the smallest positive integer n for which there exists an injective function f : V (G) → {0, 1,. .. , n} such that each uv ∈ E (G) is labeled |f (u) − f (v)| and the resulting set of edge labels is {c, c + 1,. .. , c + |E (G)| − 1} for some positive integer c or +∞ if there exists no such integer n. If c = 1, then the resulting beta-number is called the strong beta-number of G and is denoted by β s (G). In this paper, we show that if G is a bipartite graph and m is odd, then β (mG) ≤ mβ (G) + m − 1. This leads us to conclude that β (mG) = m |V (G)| − 1 if G has the additional property that G is a graceful nontrivial Full PDF DMGT Page tree. In addition to these, we examine the (strong) beta-number of forests whose components are isomorphic to either paths or stars.

The consecutively super edge-magic deficiency of graphs and related concepts

“…We conclude this paper with some remarks on the conjectures stated in [10,16] and three new conjectures.…”

“…The authors conjectured in [16] that if F is a forest of order p, then grac (F ) is either p − 1 or p. It follows from Lemma 2.1 that all the results on alpha-numbers of forests obtained in the last section validate this conjecture. Moreover, as we mentioned in the beginning of the last section, µ c (F ) is finite for any forest F .…”

“…The authors conjectured in [16] that β (F ) ≤ p and β s (F ) ≤ p for every forest F of order p. Of course, if Conjecture 1 is true, then it follows from Theorem 2.1 that α (F ) ≤ p for every forest F of order p. Indeed, it follows from Lemma 2.1 that if Conjecture 1 is true, so are the aforementioned conjectures on beta-number and strong beta-number. Moreover, by Lemma 1.4, the truth of Conjecture 3 implies the truth of the aformentioned conjecture by Figueroa-Centeno et al [10].…”

“…Next, to see that f (u) + f (v) + f (uv) = 9m/2 + 3 for all uv ∈ E (F ), notice that {f (u) + f (v) : uv ∈ E (F )} = [3m/2 + 2, 5m/2 + 1]or, equivalently,{f (u) + f (v) : uv ∈ E (F )} = [s, s + |E (F )| − 1] ,where s = 3m/2 + 2. Consequently, we conclude that µ c (F ) ≤ 1 when m is even, completing the proof.For every positive integer m, it is known from[16] that β (mP 2 ) = 2m when m ≡ 2 (mod 4) and β (mP 2 ) = 2m − 1 in all other cases. By letting G ∼ = mP 2 in Theorem 2.1, we now obtain the following result from Theorem 4.1.…”

On the strong beta-number of galaxies with three and four components

“…In [4] the authors determined the exact values for the (strong) beta-number for several classes of graphs including galaxies with two components, and proved that every nontrivial tree and forest has finite strong beta-number. Ichishima et al [5] studied the (strong) beta-number for forests with isomorphic components. This led them to conjecture that the (strong) beta-number and gracefulness of a forest of order p are either p − 1 or p .…”