Complete characterization of graphs with local total antimagic chromatic number 3

A total labeling of a graph G = (V, E) is said to be local total antimagic if it is a bijection f: V∪ E →{1,… ,|V|+|E|} such that adjacent vertices, adjacent edges, and incident vertex and edge have distinct induced weights where the induced weight of a vertex v, w_f(v) = ∑ f(e) with e ranging over all the edges incident to v, and the induced weight of an edge uv is w_f(uv) = f(u) + f(v). The local total antimagic chromatic number of G, denoted by χ_lt(G), is the minimum number of distinct induced vertex and edge weights over all local total antimagic labelings of G. In this paper, we first obtained general lower and upper bounds for χ_lt(G) and sufficient conditions to construct a graph H with k pendant edges and χ_lt(H) ∈{Δ(H)+1, k+1}. We then completely characterized graphs G with χ_lt(G)=3. Many families of (disconnected) graphs H with k pendant edges and χ_lt(H) ∈{Δ(H)+1, k+1} are also obtained.