Equitable vertex arboricity of planar graphs

Let $G_1$ be a planar graph such that all cycles of length at most 4 are independent and let $G_2$ be a planar graph without 3-cycles and adjacent 4-cycles. It is proved that the set of vertices of $G_1$ and $G_2$ can be equitably partitioned into $t$ subsets for every $t\geq 3$ so that each subset induces a forest. These results partially confirm a conjecture of Wu, Zhang and Li.

💡 Research Summary

The paper investigates the equitable vertex arboricity of planar graphs, a parameter that measures how the vertex set of a graph can be partitioned into a given number of subsets, each of which induces a forest, while the subsets are required to be as equal in size as possible. Formally, the equitable vertex arboricity a_eq(G) is the smallest integer k such that the vertices of G can be equitably partitioned into k parts each inducing a forest. The strong version a*_eq(G) is the smallest integer t such that G admits an equitable t′‑tree‑coloring for every t′ ≥ t. Wu, Zhang, and Li introduced these notions and conjectured two statements: (1) a*_eq(G) ≤ ⌈Δ(G)+½⌉ for any graph G, and (2) there exists a universal constant ζ such that a*_eq(G) ≤ ζ for every planar graph G. They proved the second conjecture for planar graphs of girth at least five, establishing a*_eq(G) ≤ 3 in that case.

The present work extends the result to two broader families of planar graphs. The first family G₁ consists of planar graphs in which every cycle of length at most four is independent, i.e., no two such cycles share a vertex. The second family G₂ comprises planar graphs without triangles and without adjacent 4‑cycles. For both families the authors prove that a*_eq(G) ≤ 3, meaning that for any integer t ≥ 3 the vertex set can be equitably split into t parts each inducing a forest.

The central technical tool is Lemma 3 (originally due to Wu, Zhang, and Li). It states that if one can select t distinct vertices x₁,…,x_t such that each vertex x_i has at most 2i−1 neighbours outside the selected set, and if the graph obtained by deleting these t vertices already has an equitable t‑tree‑coloring, then the original graph also has an equitable t‑tree‑coloring. Consequently, the authors aim to construct such a set S = {x₁,…,x_t} in a minimal counterexample and then derive a contradiction.

First, Lemma 4 shows that any graph in G₁ must have minimum degree δ(G) ≤ 3. This is proved by a discharging argument based on Euler’s formula: assigning an initial charge of d(v)−4 to each vertex and face, then redistributing charge from 5⁺‑faces to adjacent 3‑faces. The resulting contradiction forces the existence of a low‑degree vertex.

Next, a series of structural propositions (Propositions 1–7) are established for G₁. They restrict how low‑degree vertices can be adjacent to higher‑degree ones: a 2‑vertex can only be adjacent to vertices of degree at least 7; a 3‑vertex must be adjacent either to three vertices of degree at least 5 or to one 4‑vertex together with two vertices of degree at least 7; similar constraints are given for faces incident with low‑degree vertices. These propositions guarantee that when vertices are removed iteratively (as required by Lemma 3) there will always be a vertex of degree at most three in the remaining graph, ensuring the condition |N(x_i) \ S| ≤ 2i−1.

With these structural facts, the authors assume a minimal counterexample G and construct the set S by repeatedly picking a vertex of degree at most three in the current reduced graph. By minimality, G−S admits an equitable t‑tree‑coloring for any t ≥ 3, and Lemma 3 then yields an equitable t‑tree‑coloring of G, contradicting the assumption. Hence a*_eq(G₁) ≤ 3.

To complete the proof, a second discharging phase is introduced. Initial charges are set to c(v)=3d(v)−10 for vertices and c(f)=2d(f)−10 for faces, giving a total charge of –20. Five discharging rules (R1–R5) move charge from higher‑degree vertices and larger faces to low‑degree vertices and small faces. Detailed case analysis shows that after discharging every element ends with non‑negative charge, contradicting the negative total. This reinforces the impossibility of a minimal counterexample.

For the second family G₂, the authors repeat the overall strategy but need additional propositions (Propositions 8–9) because the absence of adjacent 4‑cycles imposes stronger restrictions on how many 2‑vertices can be incident to a given vertex of degree up to 13. The same Lemma 3 construction works, and the discharging argument is adapted with the same initial charge scheme and rules, now using the refined structural bounds. The authors verify that G₂ also admits equitable t‑tree‑colorings for all t ≥ 3, establishing a*_eq(G₂) ≤ 3.

In summary, the paper confirms that for both considered classes of planar graphs the strong equitable vertex arboricity is at most three. This extends the known result for planar graphs of girth at least five and provides further evidence for Wu, Zhang, and Li’s conjecture that a universal constant (indeed 3) bounds the equitable vertex arboricity of all planar graphs. The work showcases the power of combining structural decomposition (low‑degree adjacency constraints) with careful discharging arguments to resolve equitable coloring problems. Future directions may include removing the independence condition on short cycles or handling planar graphs with adjacent 4‑cycles, moving closer to a full proof of the conjecture for all planar graphs.