On the Strength and Independence Number of Graphs

A numbering f of a graph G of order n is a labeling that assigns distinct elements of the set { 1 , 2 , . . . , n } to the vertices of G . The strength str f ( G ) of a numbering f : V ( G ) → { 1 , 2 , . . . , n } of G is defined by str f ( G ) = max { f ( u ) + f ( v ) | uv ∈ E ( G ) } , that is, str f ( G ) is the maximum edge label of G and the strength str ( G ) of a graph G itself is the minimum of the set { str f ( G ) | f is a numbering of G } . In this paper, we present a necessary and sufficient condition for the strength of a graph G of order n to meet the constraints str ( G ) = 2 n − 2 β ( G ) + 1 and str ( G ) = n + δ ( G ) = 2 n − 2 β ( G ) + 1 , where β ( G ) and δ ( G ) denote the independence number and the minimum degree of G , respectively. This answers open problems posed by Gao, Lau, and Shiu [ Symmetry 13 (2021) #513]. Also, an earlier result leads us to determine a formula for the strength of graphs containing a particular class of graphs as a subgraph. We also extend what is known in the literature about k -stable properties.