Total Coloring Conjecture (general)

Establish that every finite, simple graph G admits a total coloring using at most Δ(G)+2 colors. A total coloring is a mapping φ: V(G)∪E(G) → C such that φ restricted to V(G) is a proper vertex coloring, φ restricted to E(G) is a proper edge coloring, and for every edge uv ∈ E(G), φ(uv) ≠ φ(u) and φ(uv) ≠ φ(v).

Background

The Total Coloring Conjecture, posed independently by Vizing and Behzad, is a central open problem in graph coloring. It asserts that two more than the maximum degree suffices for a total coloring of any simple graph.

The conjecture is proved for graphs of maximum degree at most 5 and for various special classes, and short proofs are known for 3-degenerate graphs. However, the conjecture remains open in general and for certain subclasses such as 4-degenerate graphs, as noted in this paper.