The total disjoint irregularity strength of some certain graphs - Meilin I Tilukay; A. N. M. Salman (Indonesian Combinatorial Society (InaCombS), 2020)
Under a totally irregular total k-labeling of a graph G = (V,E), we found that for some certain graphs, the edge-weight set W(E) and the vertex-weight set W(V) of G which are induced by k = ts(G), W(E) ∩ W(V) is a non empty set. For which k, a graph G has a totally irregular total labeling if W(E) ∩ W(V) = ∅? We introduce the total disjoint irregularity strength, denoted by ds(G), as the minimum value k where this condition satisfied. We provide the lower bound of ds(G) and determine the total disjoint irregularity strength of cycles, paths, stars, and complete graphs.