Albertson–Berman Conjecture in Planar Graphs

• The Albertson–Berman Conjecture is a fundamental graph theory topic that asserts every n-vertex planar graph contains an induced forest with at least n/2 vertices.
• Research leveraging acyclic 5-colorings and multigraph reductions has provided partial results, with current bounds ensuring induced forests of at least 0.4n vertices in some cases.
• This conjecture drives further exploration into combinatorial proofs, algorithmic complexity, and structural insights in planar and cubic graphs.

The Albertson–Berman Conjecture is a fundamental open problem in structural graph theory concerning the existence and maximal size of induced forests in planar graphs. Specifically, it posits that every planar graph on $n$ vertices contains a large induced forest, with important implications for independence numbers, coloring, and extremal combinatorics. The conjecture has prompted extensive study, resulting in related partial results, algorithmic challenges, and structural insights, especially regarding simple, bipartite, and multigraph settings.

1. Formal Statement and Motivation

Let $G$ be a simple planar graph with $n$ vertices. Define $a(G)$ as the cardinality of the largest induced forest in $G$: $$a(G) = \max \{|V(F)| : F \subseteq G\ \text{is an induced forest}\}.$$ The Albertson–Berman Conjecture asserts: $a(G) \geq \frac{n}{2}$ for all simple planar graphs $G$ (Makarov, 8 Jan 2026). This conjecture, posed in 1979, seeks a tight lower bound for the size of the largest induced forest, extending beyond trivial edgeless forests (i.e., independent sets) and linking the problem to more refined decompositional and coloring properties of planar graphs.

2. Historical Context and Known Results

The conjecture arises within the lineage of results connecting induced substructures and coloring. By elementary reasoning, $a(G) \geq \alpha(G)$, where $\alpha(G)$ is the independence number, as an independent set is a trivial induced forest (Makarov, 8 Jan 2026). The Four-Color Theorem implies $\alpha(G) \geq n/4$ for planar graphs, yielding $a(G) \geq n/4$ as a universal lower bound—no purely combinatorial (i.e., non–Four-Color–Theorem) proof of this minimum is currently known.

In 1979, Borodin demonstrated by acyclic 5-coloring that every simple planar graph can be colored with four colors such that every two color classes induce a forest. Consequently, for such a coloring, the union of the two largest color classes forms an induced forest of cardinality at least $2n/5$, thus $a(G) \geq 0.4n$ (Makarov, 8 Jan 2026). Nevertheless, the $n/2$ bound remains out of reach, even under additional constraints such as large girth.

3. Extensions: Multigraphs, Parallel Edges, and Reductions

Makarov’s work generalizes the problem to planar multigraphs (loop-free graphs potentially with parallel edges) and establishes a precise equivalence with the independence number in simple planar graphs (Makarov, 8 Jan 2026). Let $M$ be a planar multigraph on $n$ vertices. Several key reductions are as follows:

• Deduplication Lemma (2.1): Form a simple planar $G_M$ by replacing all parallel edges in $M$ with single edges. Then $a(M) \geq \alpha(G_M)$.
• Doubling Lemma (2.2): For a simple planar $G$, define $M_G$ by doubling each edge of $G$. Then $a(M_G) = \alpha(G)$.

This equivalence implies: $\max_{|V|=n} a(M) = \max_{|V|=n} \alpha(G),$ so bounds for $\alpha(G)$ translate directly to $a(M)$, with the Four-Color Theorem again supplying $a(M) \geq n/4$. This bound is tight: doubling all edges in the disjoint union of $k$ copies of $K_4$ yields $a(M) = n/4$ for $n = 4k$.

Further refinements occur when the number $k$ of vertex pairs joined by parallel edges is small:

• Subdivision-based Reduction: Subdividing one copy of each parallel edge in $M$ produces a simple planar $G_M$ with $n+k$ vertices and $a(G_M) = a(M) + k$. Applying the $n/2$ conjecture yields $a(M) \geq (n-k)/2$ if the conjecture holds.
• Acyclic 5-Coloring Argument: Direct application to the deduplicated graph gives $a(M) \geq \frac{2}{5}n - \frac{k}{10}$, improving the trivial bound when $k$ is small.

In scenarios where $M$ is a planar multigraph without any 2-faces (i.e., embeddings where no face is a digon), even stronger results emerge. A discharging argument yields: $a(M) \geq \frac{3}{10} n + \frac{7}{30}.$ The construction of graphs achieving $a(M) = (3/7)n + (4/7)$ demonstrates the near-optimality of this bound (Makarov, 8 Jan 2026).

4. Connections with Spanning Trees: Hists and Related Conjectures

In parallel, the study of homeomorphically irreducible spanning trees (Hists), i.e., spanning trees without degree-2 vertices, features a conjecture originally advanced by Albertson and Berman in a different context (Hoffmann-Ostenhof et al., 2015). For a connected cubic graph $G$, a Hist $T \subset G$ has only degree-1 and degree-3 vertices. A key condition is that the complement $H = \langle E(G) \setminus E(T)\rangle$ is a non-separating 2-regular subgraph with $|V(H)| = |V(G)|/2 + 1$.

This led to the following conjecture:

For every positive integer $k$, there exists a cyclically $k$-edge-connected cubic graph $G$ that does not admit any Hist.

The conjecture was settled affirmatively by constructing, for every $k\geq 1$, a cyclically $k$-edge-connected bipartite cubic graph $G$ with $|V(G)| \equiv 0 \pmod 4$ (and thus $G$ cannot admit a Hist). The argument proceeds via inflations of highly connected regular graphs, a method using both probabilistic and explicit constructions (Hoffmann-Ostenhof et al., 2015).

5. Illustrative Examples and Boundary Cases

Several explicit examples delineate the sharpness of the conjecture and its related propositions:

• For $G = K_{3,3}$ (bipartite cubic, six vertices), $a(G) = 6 \geq 3$, showing the $|V| \equiv 2 \pmod 4$ condition for Hists is necessary but not sufficient.
• The family of doubled $K_4$ multigraphs $M_k$ demonstrates tightness for $a(M) \geq n/4$ in the presence of parallel edges.
• In the 2-face-free setting, an explicit family of multigraphs $M_k$ with $n_k = 7k+1$ and $a(M_k) = 3k+1$ achieves $a(M_k) = (3/7)n_k + (4/7)$, showing that more ambitious bounds cannot hold in this regime (Makarov, 8 Jan 2026).

6. Broader Impact and Open Directions

The Albertson–Berman Conjecture anchors a significant direction in graph theory, linking chromatic, independence, and induced subgraph parameters. Some prominent open issues and research avenues include:

• Strengthening Bounds: Seek combinatorial proofs or improvements for $a(G) \geq n/2$ in specific subclasses (e.g., higher girth, bipartite, forbidden minor conditions).
• Algorithmic Complexity: Deciding the existence or size of large induced forests, and especially Hists, in planar or cubic graphs remains NP-complete in general, motivating parameterized and approximation frameworks (Hoffmann-Ostenhof et al., 2015).
• Extensions to Multigraphs: Determine for which additional forbidden configurations (such as small faces) the induced forest bounds can be further increased.
• Enumeration: Quantify how many graphs, within families such as fullerenes and polyhedral cages, admit large induced forests or Hists.
• Interplay of Structural Properties: Prospective work includes generalizing the sharp parity and half-vertex obstructions from regular graphs and exploring extremal phenomena relating cyclic connectivity, girth, and admissible spanning-tree patterns (Hoffmann-Ostenhof et al., 2015).

The conjecture and its refinements thus unify combinatorial, algebraic, and algorithmic aspects, with direct applications to coloring, polyhedral graph embeddings, and extremal planar graph theory.