A Result on the Strength of Graphs by Factorizations of Complete Graphs

Abstract

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 of G is defined by str (G) = min {strf (G) |f is a numbering of G} , where strf (G) = max {f (u) + f (v) |uv ∈ E (G)}. In this paper, we present some results obtained from factorizations of complete graphs. In particular, we show that for every k ∈ [1, n− 1], there exists a graph G of order n satisfying δ (G) = k and str (G) = n+ k, where δ (G) denotes the minimum degree of G.