On Generalized Total Colourings of Planar Graphs

Abstract: In this paper we study generalised total colourings of graphs where the colour classes formed by vertices and edges, respectively, induce forests, while incident edges/vertices receive distinct colours. In [M. Borowiecki and I. Broere, Hamiltonicity and Generalised Total Colourings of Planar Graphs, Discussiones Mathematicae Graph Theory 36 (2016) 243--257] it was conjectured that for planar graphs, four colours suffice for this type of colouring. We confirm this conjecture for two infinite families of planar graphs.

• The paper constructs two infinite families of hamiltonian maximal planar graphs and verifies that the conjectured bound of 4 colours for generalized total colouring is achieved.
• Two families of graphs, MH and triangulated square grids are constructed to validate the conjecture stating that for each graph vertical 带 1 lead to a hamiltonian maximal planar graph and achieve the 4-colour bound.
• The evidence strongly supports the generalized total coloring ±1, where edges must induce forests continued in colour.

This paper studies $({\cal D}_1, {\cal D}_1)$-total colourings of planar graphs — colourings of vertices and edges in which each vertex colour class induces a forest, each edge colour class induces a forest, and incident vertices and edges receive distinct colours. The central object is a conjecture attributed to Borowiecki and Broere (2608.19294), stating that every planar graph $G$ satisfies $\chi''_{{\cal D}_1,{\cal D}_1}(G) \le 4$. The paper does not settle the conjecture in full; rather, it constructs two infinite families of hamiltonian maximal planar graphs and proves that the conjectured bound holds with equality on both.

Background: generalized total colourings via hereditary properties

The framework is the $( {\cal P}, {\cal Q})$-total colouring of Borowiecki, Kemnitz and Mihók, where ${\cal P}$ and ${\cal Q}$ are additive hereditary graph properties. Vertices coloured alike must induce a subgraph in ${\cal P}$, edges coloured alike must induce a subgraph in ${\cal Q}$, and properness is enforced only between incident vertices and edges. Here ${\cal D}_1$ denotes the class of graphs whose components are trees (acyclic graphs), so the colourings considered require both vertex and edge colour classes to induce forests. This generalizes classical total chromatic number (the case ${\cal P} = {\cal O}$, $G$0) studied since Behzad and Vijayaditya.

Prior work established that even planar triangulations admitting a Hamilton cycle with no cycle among interior edges admit a 4-colour such colouring, motivating the conjecture for all planar graphs. Both families constructed here are hamiltonian, consistent with the pattern noted by the authors that essentially all known supporting evidence involves hamiltonian planar graphs.

First family: the class $G$1

The class $G$2 consists of graphs built from an $G$3-cycle $G$4 ($G$5) by adding chords inside the cycle connecting pairs whose indices sum to $G$6 or $G$7, and, for $G$8, edges from $G$9 to most other vertices outside the cycle. A counting argument shows these graphs have size exactly $\chi''_{{\cal D}_1,{\cal D}_1}(G) \le 4$0, hence are maximal planar by the standard characterization, and they are manifestly hamiltonian.

The upper bound proof proceeds in two stages. First, an explicit two-colouring $\chi''_{{\cal D}_1,{\cal D}_1}(G) \le 4$1 is defined using residues modulo 3 ($\chi''_{{\cal D}_1,{\cal D}_1}(G) \le 4$2): $\chi''_{{\cal D}_1,{\cal D}_1}(G) \le 4$3 receives colour 1, vertices $\chi''_{{\cal D}_1,{\cal D}_1}(G) \le 4$4 with $\chi''_{{\cal D}_1,{\cal D}_1}(G) \le 4$5 up to $\chi''_{{\cal D}_1,{\cal D}_1}(G) \le 4$6, together with their reflections $\chi''_{{\cal D}_1,{\cal D}_1}(G) \le 4$7, receive colour 1, and all remaining vertices receive colour 2. A direct structural argument shows colour class $\chi''_{{\cal D}_1,{\cal D}_1}(G) \le 4$8 induces a path while $\chi''_{{\cal D}_1,{\cal D}_1}(G) \le 4$9 induces an independent set apart from $( {\cal P}, {\cal Q})$0; both are forests.

Second, deleting all monochromatic edges yields a spanning subgraph $( {\cal P}, {\cal Q})$1 which is shown to be edge-decomposable into two forests. The key lemma is a nested structure property: removing a particular degree-2 vertex from $( {\cal P}, {\cal Q})$2 leaves a graph isomorphic to $( {\cal P}, {\cal Q})$3 for the corresponding smaller member of $( {\cal P}, {\cal Q})$4. This enables a greedy peeling algorithm that distributes the two edges incident with each degree-2 vertex between two forest copies $( {\cal P}, {\cal Q})$5, terminating at $( {\cal P}, {\cal Q})$6. Each added edge attaches at a vertex never revisited within its assigned copy, so acyclicity holds by construction.

Edges removed before forming $( {\cal P}, {\cal Q})$7 (i.e., monochromatic edges under $( {\cal P}, {\cal Q})$8) are recoloured with the opposite vertex colour, giving a valid total colouring with four colours.

For the lower bound, the authors observe $( {\cal P}, {\cal Q})$9 for every ${\cal P}$0 and prove ${\cal P}$1 by exhaustive case analysis: with two vertex colours, an alternatingly-coloured 4-cycle forces four edge colours; with three vertex colours, the remaining five edges form two triangles sharing an edge, and any 3-edge-colouring creates a monochromatic triangle or forces a fourth colour. Hence:

${\cal P}$2

Since ${\cal P}$3 is monotone under subgraphs, every planar subgraph of a graph in ${\cal P}$4 also admits a 4-colouring. Note that this establishes tightness of the conjecture's bound on these families: three colours provably do not suffice.

Second family: triangulated square grids

The second construction takes an ${\cal P}$5 grid (${\cal P}$6), adds one diagonal in each unit 4-cycle (edges ${\cal P}$7), and joins ${\cal P}$8 to all boundary vertices except its grid neighbours. A size computation gives exactly ${\cal P}$9 edges, so the resulting graph ${\cal Q}$0 is again maximal planar and clearly hamiltonian.

The upper-bound argument mirrors the first but with different structure. The vertex colouring ${\cal Q}$1 partitions ${\cal Q}$2 so that ${\cal Q}$3 induces a generalized star centered at ${\cal Q}$4 with ${\cal Q}$5 arms, and ${\cal Q}$6 induces a generalized caterpillar with spine ${\cal Q}$7 of length ${\cal Q}$8. After deleting monochromatic edges, the residual spanning subgraph ${\cal Q}$9 splits into two explicitly defined edge sets ${\cal P}$0: ${\cal P}$1 is a generalized star at ${\cal P}$2, and ${\cal P}$3 is a union of a generalized star and paths. Colouring ${\cal P}$4's edges with colour 2, ${\cal P}$5's edges with colour 1, ${\cal P}$6 with colour 3 and ${\cal P}$7 with colour 4 completes the 4-colouring.

For the lower bound, the authors exhibit a wheel ${\cal P}$8 of order 5 as an induced subgraph of every triangulated grid (on ${\cal P}$9). A three-case analysis shows no ${\cal Q}$0-total colouring of ${\cal Q}$1 uses only three colours: alternating vertex colours force four colours on the rim; adjacent-equal patterns force either a monochromatic rim or a contradiction at the hub; and three-of-a-kind patterns force a monochromatic triangle or 4-cycle regardless of the hub's colour. Consequently ${\cal Q}$2 for every triangulated square grid, and by monotonicity every planar subgraph of one also satisfies the 4-colour bound.

A minor presentational issue arises here: the corollary states the subgraph bound using the symbol ${\cal Q}$3 where ${\cal Q}$4 is intended, though the claim itself follows directly from subgraph monotonicity.

Limitations and open questions

The paper confirms the 4-colour conjecture only on the two constructed families, both of which are maximal planar and hamiltonian. Several gaps remain explicit:

• Non-hamiltonian planar graphs: no evidence is provided beyond hamiltonian examples, despite non-hamiltonian planar triangulations being well known; extending the method to such graphs is left open.
• Non-maximal planar graphs: the constructions rely heavily on maximal planarity (via the ${\cal Q}$5 edge count and triangulation structure); sparse planar graphs are not addressed.
• General planar graphs: the full conjecture ${\cal Q}$6 for arbitrary planar ${\cal Q}$7 remains unproven.
• Other property pairs: results concern only ${\cal Q}$8; the behaviour of related pairs such as ${\cal Q}$9 on these same families is not treated here.

Conclusion

The paper verifies the conjectured bound ${\cal D}_1$0 with equality on two infinite classes of hamiltonian maximal planar graphs — the chord-completion family ${\cal D}_1$1 and triangulated square grids. Both proofs follow a common template: a structured two-colouring of the vertices into induced forests, followed by a decomposition of the bichromatic remainder into two edge-induced forests, yielding four colours in total, with matching lower bounds supplied by ${\cal D}_1$2 and the 5-wheel respectively. By subgraph monotonicity, all planar subgraphs of members of these families inherit the 4-colour bound. The general conjecture for all planar graphs, particularly those without Hamilton cycles, remains open.