The total colouring conjecture (Behzad 1965, Vizing 1968)

The total colouring conjecture (Behzad 1965, Vizing 1968). Every finite simple graph G has a total colouring with at most Δ(G) + 2 colours, i.e. χ''(G) ≤ Δ(G) + 2, where Δ(G) is the maximum degree.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-total-coloring-conjecture,
  title        = {The total colouring conjecture (Behzad 1965, Vizing 1968)},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/total-coloring-conjecture}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

The total colouring conjecture (Behzad 1965, Vizing 1968).

Every finite simple graph $G$ has a total colouring with at most $Δ (G) + 2$ colours, i.e. $χ^{''} (G) \leq Δ (G) + 2$ , where $Δ (G)$ is the maximum degree.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.TotalColoringConjecture.

theorem total_coloring_conjecture :
    ∀ {V : Type} [Fintype V] [DecidableEq V] (G : SimpleGraph V) [DecidableRel G.Adj],
      G.totalChromaticNumber ≤ G.maxDegree + 2

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
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A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Wikipedia
[Be65] Behzad, M. (1965). Graphs and their chromatic numbers. Ph.D. thesis, Michigan State University.
[Vi68] Vizing, V. G. (1968). "Some unsolved problems in graph theory." Uspekhi Mat. Nauk 23, pp. 117--134.
[Ko96] Kostochka, A. V. (1996). "The total chromatic number of any multigraph with maximum degree five is at most seven." Discrete Math. 162, pp. 199--214.
[MR98] Molloy, M. and Reed, B. (1998). "A bound on the total chromatic number." Combinatorica 18, pp. 241--280.
[Yap96] Yap, H. P. (1996). Total colourings of graphs. Lecture Notes in Mathematics 1623, Springer.

Source and licence

Imported from Formal Conjectures (Wikipedia), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.