On inclusive and non-inclusive vertex irregular d-distance vertex labeling

Abstract

In this paper, we generalise the notion of distance irregular labeling introduced by Slamin to vertex irregular \(d\)-distance vertex labeling, for any distance \(d\) up to the diameter. We also define the inclusive vertex irregular \(d\)-distance vertex labeling. We give the lower bound of the inclusive vertex irregular \(1\)-distance vertex labeling for general graphs and a better lower bound on caterpillars. The inclusive labelings for paths \(P_n, n \equiv 0 \mod 3\), stars \(S_n\), double stars \(S(m,n)\), cycles \(C_n\), and wheels \(W_n\) are provided. From the inclusive vertex irregular \(1\)-distance vertex labeling on cycles, we derive the vertex irregular \(1\)-distance vertex labeling on prisms.