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

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 \(d\geq2\).