Extremal values for a γ-labeling of a cycle with a triangle

Abstract

For any graph G of order n and size m, a γ-labeling of G is defined as a one-to-one function f: V(G) → {0, 1, ..., m} that induces an edge-labeling f′: E(G) → {1, 2, ..., m} on G defined by f′(uv) = |f(u) - f(v)|, for each edge uv in E(G). The value of f is defined by val(f) = Σ_eεE(G) f′ (e). In this paper, we determine the extremal values of a γ-labeling of a cycle with a triangle. Several examples are considered. More generally, for a nonnegative integer δ, we define a γ^δ-labeling of G as a one-to-one function f : V(G) → {0, 1, ..., m+δ-1, m+δ} that induces an edge-labeling f′: E(G) → {1, 2, ..., m + δ} on G defined by f′(uv) = |f(u) - f(v)|, for each edge uv in E(G). The value of a γ^δ-labeling f is defined by val(f) = Σ_eεE(G) f′ (e). We determine the extremal values of a γ^δ-labeling of some graphs.