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 $P_n,n\equiv 0\mathrm{mod}3$, 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.