Hulgan’s 5-color AVD total coloring conjecture for graphs with maximum degree 3

Show that every finite, simple, undirected graph G with maximum degree Δ(G)=3 admits an adjacent-vertex-distinguishing total coloring using 5 colors.

Background

For graphs with maximum degree 3, the AVD total coloring conjecture (Δ+3 bound) is known to hold, but Hulgan proposed a stronger bound of 5 colors (which equals Δ+2 for Δ=3).

The paper notes necessary lower bounds (e.g., graphs with adjacent maximum-degree vertices require at least Δ+2 colors) and records Hulgan’s conjecture as open for Δ=3 graphs.