On (k, λ)-magically total labeling of graphs

摘要

If there are integers k and λ ≠ 0 such that a total labeling / of a connected graph G = (V,E) from V ∪ E to {1,2,..., |V| + |E|} satisfies f(x) ≠ f(y) for distinct x,y ∈ V ∪ E and f(u) + f(v) = k + λf(uv) for each edge uv ∈ E, then f is called a (k, λ)-magically total labeling ((k, λ)-mtl for short) of G. Several properties of (k, λ)-mtls of graphs are shown. The sufficient and necessary connections between (k,λ)-mtls and several known labelings (such as graceful, odd-graceful, felicitous and (b,d)-edge antimagic total labelings) are given. Furthermore, every tree is proven to be a subgraph of a tree having super (k, λ)-mtls.