AVD-total-chromatic number of some families of graphs with Δ(G)=3

An AVD-total-colouring of a simple graph G is a mapping π:V(G)∪E(G)→{1,…,k}, k≥1, such that: (i) for each pair of adjacent or incident elements x,y∈V(G)∪E(G), π(x)≠π(y); and (ii) for each pair of adjacent vertices x,y∈V(G), sets {π(x)}∪{π(xv):xv∈E(G),v∈V(G)} and {π(y)}∪{π(yv):yv∈E(G),v∈V(G)} are distinct. The AVD-total-chromatic number, χa″(G), is the smallest number of colours for which G admits an AVD-total-colouring. In 2010, J. Hulgan conjectured that any simple graph G with maximum degree three has χa″(G)≤5. In this article, we verify Hulgan's Conjecture for simple graphs G with Δ(G)=3 and without adjacent vertices of maximum degree, and also for the following families of snarks: the flower snarks, generalized Blanuša snarks, and LP1-snarks. In fact, we determine the exact value of χa″(G) for all families considered in this work.