Four-colour conjecture for generalized total colourings of planar graphs

Determine whether every planar graph G satisfies the generalized total chromatic bound \(\chi''_{\mathcal{D}_1,\mathcal{D}_1}(G) \leq 4\), where each vertex colour class and each edge colour class induces a forest and incident vertices and edges receive distinct colours.

Background

The paper studies generalized total colourings in which both monochromatic vertex subgraphs and monochromatic edge subgraphs are forests; the corresponding parameter is $\chi''_{\mathcal{D}_1,\mathcal{D}_1}(G)$. A conjecture from the cited earlier work asserts that four colours suffice for every planar graph under these constraints.

The paper proves the bound, with equality, for two infinite families of planar graphs: the class $\mathsf{MH}$ of constructed hamiltonian maximal planar graphs and the family of triangulated square grids. It therefore confirms the conjecture only for these families, leaving the assertion for arbitrary planar graphs unresolved.