Albertson–Berman Conjecture

• The Albertson–Berman Conjecture is a statement in graph theory asserting that every simple planar graph contains an induced forest on at least half of its vertices.
• Researchers utilize reductions to independence numbers, acyclic colorings, and discharging techniques to derive effective bounds and structural insights.
• The conjecture’s framework extends to planar multigraphs and cubic graphs, highlighting its key role in advancing extremal combinatorics and related open problems.

The Albertson–Berman Conjecture concerns the existence and quantitative lower bounds of large induced forests in planar graphs, a problem central to extremal combinatorics and graph theory. It postulates that every simple planar graph on $n$ vertices contains an induced forest on at least $n/2$ vertices. Despite substantial research efforts since its proposal in 1979, this conjecture remains open for simple planar graphs. The conjecture's interaction with related problems, such as its extension to planar multigraphs and the existence of special types of spanning trees in cubic graphs, has been explored through various structural, coloring, and combinatorial methods.

1. Formulation of the Albertson–Berman Conjecture

The Albertson–Berman Conjecture is formally stated as follows:

For every simple planar graph $G$ on $n$ vertices, the number of vertices in its largest induced forest satisfies $a(G)\;\ge\;\frac n2,$ where $a(G)$ is the size of the largest induced forest in $G$ (Makarov, 8 Jan 2026).

This conjecture targets an extremal property of simple planar graphs, focusing on the minimal possible size of induced acyclic subgraphs as a fraction of the total number of vertices. No counterexamples are known, and the best general lower bound presently is $a(G)\ge n/4$, derived via the Four-Color Theorem and the independence number (Makarov, 8 Jan 2026).

2. Connections to Planar Multigraphs and Independence Numbers

Allowing parallel edges leads to a natural extension of the conjecture to planar multigraphs. The study in "Large induced forests in planar multigraphs" establishes that the induced forest problem for planar multigraphs is reducible to the independence number problem for simple planar graphs.

Key reductions include:

• For any planar multigraph $M$, one can produce a simple planar graph $G_M$ with the same vertex set such that

$n/2$0

where $n/2$1 is the independence number of $n/2$2 (Lemma 2.1). By deleting parallel edges in $n/2$3, any independent set in $n/2$4 yields an induced forest in $n/2$5 (Makarov, 8 Jan 2026).

• Conversely, duplicating each edge in a simple planar graph $n/2$6 to form $n/2$7 results in

$n/2$8

because every induced forest in $n/2$9 must avoid the endpoints of duplicate edges, enforcing independence in $G$0 (Lemma 2.2) (Makarov, 8 Jan 2026).

This correspondence sharpens the focus on independence numbers as the central quantity governing induced forests in the multigraph context.

3. Best-Known Bounds and Tightness in Planar Multigraphs

The study of induced forests extends to bounding $G$1 for planar multigraphs. Theorem 2.3 in (Makarov, 8 Jan 2026) establishes

$G$2

which is tight. The tightness is witnessed by constructing a disjoint union of $G$3 subgraphs and duplicating every edge; such configurations attain the bound.

When restricting attention to planar multigraphs with a small number $G$4 of parallel-edge pairs, more refined bounds can be obtained. Theorem 4.7 provides

$G$5

using acyclic 5-colorings and careful averaging over color class unions. This approach leverages Borodin’s theorem on acyclic colorings, partitioning the graph and managing the interaction between parallel-edge pairs and color classes, thereby improving the bound as $G$6 decreases (Makarov, 8 Jan 2026).

Assuming the Albertson–Berman Conjecture holds for simple planar graphs, an even sharper result follows: $G$7 Here a subdivision argument adds $G$8 new vertices (one per parallel-edge pair), applies the conjectured bound, and then removes the subdivisions, preserving the lower bound except for the $G$9 pairs (Theorem 4.4) (Makarov, 8 Jan 2026).

4. Induced Forests in Planar Multigraphs Without 2-Faces

A significant variant arises by prohibiting 2-faces (faces bounded by exactly two parallel edges, or “digons”) in the planar embedding. For such planar multigraphs, the bounding argument is altered due to stronger connectivity and face-degree constraints.

Theorem 5.10 asserts that for every planar multigraph $n$0 on $n$1 vertices and with no 2-faces,

$n$2

This proof employs discharging techniques, rooted forests on 2-cycles, and combinatorial charge redistribution arguments tightly coupled to the graph’s face structure. A matching upper construction (Theorem 5.12) produces an infinite family of 2-face-free planar multigraphs with

$n$3

demonstrating that the optimal constant for such graphs lies strictly between $n$4 and $n$5 (Makarov, 8 Jan 2026).

5. The Albertson–Berman Conjecture for Cubic Graphs and Homeomorphically Irreducible Spanning Trees

A related question attributed to Albertson, Berman, Hutchinson, and Thomassen concerns the existence of homeomorphically irreducible spanning trees (Hists) in cubic graphs. A spanning tree is a Hist if no vertex has degree exactly 2 in the tree.

The key results, as presented in (Hoffmann-Ostenhof et al., 2015), summarize as follows:

• Necessary Condition: In any cubic graph $n$6 admitting a Hist $n$7, the complement $n$8 of $n$9 (edges not in $a(G)\;\ge\;\frac n2,$0) forms a non-separating 2-regular subgraph, with

$a(G)\;\ge\;\frac n2,$1

For bipartite cubic graphs, this implies $a(G)\;\ge\;\frac n2,$2: thus, no bipartite cubic graph with $a(G)\;\ge\;\frac n2,$3 can have a Hist (Corollary 1.2) (Hoffmann-Ostenhof et al., 2015).

• Existential Result: For every positive integer $a(G)\;\ge\;\frac n2,$4, there exists a cyclically $a(G)\;\ge\;\frac n2,$5-edge-connected cubic graph without any Hist. The construction uses a method called inflation, replacing vertices in regular graphs by chordless cycles, to obtain high cyclic edge-connectivity and block the existence of Hists, thus settling the question posed in [ABHT] affirmatively (Hoffmann-Ostenhof et al., 2015).
• Complexity Result: Testing for the existence of a Hist in planar graphs of maximum degree 3 is NP-complete, but the aforementioned congruence obstruction offers many polynomial-time negative answers in structured cases.

6. Impact, Open Problems, and Structural Implications

The Albertson–Berman Conjecture and its relatives have stimulated substantial research on induced substructure size problems in planar and regular graphs. For planar multigraphs, the reduction to independence number and coloring arguments has unified the treatment of induced forests across simple and multigraph cases.

The main open questions include:

• Resolution of the conjecture for simple planar graphs: No counterexample is known, and the best general bound remains $a(G)\;\ge\;\frac n2,$6, while closing the gap to $a(G)\;\ge\;\frac n2,$7 is a significant goal (Makarov, 8 Jan 2026).
• Tightened constants for special classes: In the case of planar multigraphs without 2-faces, identifying the exact constant between $a(G)\;\ge\;\frac n2,$8 and $a(G)\;\ge\;\frac n2,$9 remains unresolved (Makarov, 8 Jan 2026).
• Characterization for Hists in cubic graphs: While the nonexistence for many bipartite cubic graphs is settled and highly cyclically edge-connected non-Hist graphs have been constructed, a complete characterization for cubic graphs admitting Hists is open; so are analogous questions for higher-degree regular graphs (Hoffmann-Ostenhof et al., 2015).
• Algorithmic complexity: While NP-completeness prevails in the general case, substantial graph families permit structural obstructions that aid in efficient nonexistence certification.

The conjecture’s interaction with fundamental results such as the Four-Color Theorem, acyclic colorings, and extremal subgraph theory underlines its continuing importance as an open challenge and a tool for advancing combinatorial graph theory.