(untitled) (Contemporary Mathematics, 2024)
In 2022, the result that the sum of the lengths of any two edges of a triangle is greater than the length of the third edge in Euclidean geometry, is applied to labeling graph theory, a new concept-the vertex Euclidean graph is introduced. A simple graph G = (V, E) is said to be vertex Euclidean if there exists a bijection f from V to 1, 2, ..., |V| such that f(u) + f(v) > f(w) for each C3 subgraph with vertex set u, v, w, where f(u) < f(v) < f(w). The vertex Euclidean deficiency of a graph G, denoted µvEuclid(G), is the smallest positive integer m such that G ∪Nm is vertex Euclidean. In this paper, the sufficient condition that the disjoint union of G and H is vertex Euclidean is given, meanwhile, the vertex Euclidean properties of four classes graphs are discussed, the vertex Euclidean deficiency of these graphs are obtained.