$(d,1)$-Total labellings of two infinite families of snarks

Abstract

A $(d,1)$-total labelling of a graph $G$ is an assignment of integers $\{0,1,\cdots ,l\}$ to the vertices and edges of the graph such that adjacent vertices receive distinct integers, adjacent edges receive distinct integers, and the integer received by a vertex differs at least $d$ from those received by its incident edges. The minimum number $l$ required for such an assignment is called the $(d,1)$-total number of the graph $G$. This paper contributes to $(d,1)$-total labelling of two infinite families of snarks, the Goldberg family and the Loupekhine family. We completely determine the $(d,1)$-total numbers of these two families of snarks for all $d\ge 2$.