The complexity of planar graph choosability

A graph $G$ is {\em $k$-choosable} if for every assignment of a set $S(v)$ of $k$ colors to every vertex $v$ of $G$, there is a proper coloring of $G$ that assigns to each vertex $v$ a color from $S(v)$. We consider the complexity of deciding whether a given graph is $k$-choosable for some constant $k$. In particular, it is shown that deciding whether a given planar graph is 4-choosable is NP-hard, and so is the problem of deciding whether a given planar triangle-free graph is 3-choosable. We also obtain simple constructions of a planar graph which is not 4-choosable and a planar triangle-free graph which is not 3-choosable.

💡 Research Summary

The paper investigates the computational complexity of determining whether a given graph is k‑choosable (list‑colorable with lists of size k) for fixed constants k, focusing on planar graphs and planar triangle‑free graphs. After reviewing basic definitions—k‑choosability, choice number ch(G), and known results such as every planar graph being 5‑choosable and the existence of non‑4‑choosable planar graphs—the authors present two main contributions.

First, they construct much smaller counter‑examples than previously known. Theorem 1.7 exhibits a planar graph H′ with only 75 vertices that is not 4‑choosable. H′ is built from twelve copies of a gadget W₁ (shown in Figure 1) whose two distinguished vertices u and v are identified across copies, and additional edges are added to force a specific list assignment S(u)=S(v)={7,8,9,10}. By assigning each copy a distinct ordered pair (a,b) from the 12 possible unequal pairs in S(u)×S(v), they guarantee that any attempt to color the central vertices forces an unsatisfiable coloring in at least one copy, proving non‑choosability. A second, even more compact construction (Theorem 1.8) yields a planar triangle‑free graph H′ with 164 vertices that is not 3‑choosable. This graph uses nine copies of a different gadget W₂ (Figure 2), again identifying all u’s and all v’s, and assigning S(u)={10,11,12}, S(v)={13,14,15}. By arranging the nine ordered pairs (a,b) in a cyclic order and linking successive copies through shared vertices, the authors create a forced conflict that makes any proper list‑coloring impossible.

Second, the paper establishes Π₂‑completeness for several decision problems concerning choosability. The authors start from the Restricted Planar Satisfiability (RPS) problem, a quantified Boolean formula problem with a planar incidence graph, and prove it is Π₂‑complete (Lemma 3.1). They then reduce RPS to the bipartite planar (2,3)-choosability problem (BPG(2,3)-CH), which is already known to be Π₂‑complete, by constructing a bipartite planar graph G together with a function f:V→{2,3}. The construction uses a collection of specially designed subgraphs—propagators, half‑propagators, multi‑output propagators, and initial graphs—that simulate the logical structure of the quantified formula. Each half‑propagator guarantees that the output color is the opposite of the input color, thereby encoding universal quantifiers; multi‑output propagators distribute a truth value to three clause literals; and the initial graphs provide the necessary base assignments. Because each variable appears in at most three clauses, the resulting graph remains planar.

With this reduction in hand, the authors prove Theorem 1.9 (Π₂‑completeness of BPG(2,3)-CH). They then extend the technique to the planar graph 4‑choosability problem (PG 4‑CH) and the planar triangle‑free graph 3‑choosability problem (PTFG 3‑CH). By embedding the previously constructed gadgets into planar graphs and carefully controlling the list sizes (all vertices receive lists of size 3 or 4), they show that a planar graph is 4‑choosable iff the underlying quantified formula is true, yielding Π₂‑completeness for PG 4‑CH (Theorem 1.11). An analogous argument gives Π₂‑completeness for PTFG 3‑CH (Theorem 1.10). Finally, they address the Union‑of‑Two‑Forests 3‑choosability problem (U2F 3‑CH), noting that every planar triangle‑free graph can be expressed as the union of two forests; using the same gadget constructions they prove this problem is also Π₂‑complete (Theorem 1.12).

The paper is organized as follows: Section 2 presents the small counter‑examples and proves their non‑choosability. Section 3 details the Π₂‑completeness of RPS and the reduction to BPG(2,3)-CH. Section 4 contains the reductions to PG 4‑CH, PTFG 3‑CH, and U2F 3‑CH. The authors conclude by highlighting that the choosability decision problems for planar graphs sit at the second level of the polynomial hierarchy, contrasting with the NP‑completeness of ordinary planar graph coloring, and they suggest future work on tighter bounds for minimal non‑choosable planar graphs and extensions to other graph families.