Total Coloring Conjecture for 4-degenerate graphs

Establish the Total Coloring Conjecture for 4-degenerate graphs; specifically, prove that every 4-degenerate graph G admits a total coloring using at most Δ(G)+2 colors.

Background

While short proofs of the Total Coloring Conjecture exist for 3-degenerate graphs, the conjecture remains unresolved for 4-degenerate graphs in general. This paper explicitly highlights that gap.

Degeneracy stratifies graphs by the maximum minimum degree over subgraphs, and settling the 4-degenerate case would advance understanding of the conjecture for broader sparse graph classes.