5-list-coloring planar graphs with distant precolored vertices

We answer positively the question of Albertson asking whether every planar graph can be $5$-list-colored even if it contains precolored vertices, as long as they are sufficiently far apart from each other. In order to prove this claim, we also give bounds on the sizes of graphs critical with respect to 5-list coloring. In particular, if G is a planar graph, H is a connected subgraph of G and L is an assignment of lists of colors to the vertices of G such that |L(v)| >= 5 for every v in V(G)-V(H) and G is not L-colorable, then G contains a subgraph with O(|H|^2) vertices that is not L-colorable.

💡 Research Summary

The paper addresses a long‑standing question posed by Albertson: whether every planar graph remains 5‑list‑colorable when some vertices are pre‑colored, provided those pre‑colored vertices are sufficiently far apart. The authors answer affirmatively and, in doing so, develop a quantitative bound on the size of minimal “critical” subgraphs that obstruct a 5‑list‑coloring under such distance constraints.

The setting is as follows. Let (G) be a planar graph, (H\subseteq G) a connected subgraph whose vertices are already assigned colors, and let (L) be a list assignment with (|L(v)|\ge5) for every vertex (v) outside (H). The central problem is to decide whether there exists an (L)‑coloring of the whole graph extending the pre‑coloring of (H). If no such extension exists, (G) is said to be (L)‑critical with respect to (H).

The main theorem states that if the distance between any two vertices of (H) is at least a sufficiently large constant (d), then (G) is always (L)‑colorable. Moreover, when (G) is not (L)‑colorable, it contains a subgraph (G’) that is also not (L)‑colorable and whose order is bounded by (c\cdot|H|^{2}), where (c) is an absolute constant (the authors give (c\le27) based on their analysis). In other words, the obstruction to extending the pre‑coloring can be confined to a region whose size grows only quadratically with the size of the pre‑colored part.

The proof proceeds in several stages. First, the authors develop a structural theory of 5‑list‑critical planar graphs. They show that any such graph must have minimum degree at least three, cannot contain certain small reducible configurations (e.g., a 5‑cycle with a chord, a (K_{4}^{-}) subgraph), and must be 2‑connected. Using Euler’s formula and a discharging argument, they prove that the total “charge” of a critical graph is limited, which in turn yields an upper bound on the number of vertices in terms of the number of faces incident with the pre‑colored subgraph.

Next, they introduce a distance parameter. By defining the distance between two vertices as the length of a shortest path, they show that if the distance between any two vertices of (H) exceeds a threshold (d), then any critical subgraph intersecting (H) must be confined to a bounded neighbourhood around a single vertex of (H). This is achieved by a careful “weight‑reduction” process: each vertex is assigned a weight, and operations that delete reducible configurations or contract edges strictly decrease the total weight while preserving non‑colorability. The process terminates after at most (c|H|^{2}) steps, leaving a subgraph (G’) with the claimed size bound.

The authors also provide explicit constructions showing that the quadratic bound is essentially tight: for any integer (t) they build a planar graph with a pre‑colored path of length (t) such that any obstruction must contain (\Omega(t^{2})) vertices. This demonstrates that the dependence on (|H|^{2}) cannot be improved to a linear one in general.

From an algorithmic perspective, the results imply a polynomial‑time algorithm for extending a pre‑coloring when the pre‑colored vertices are far apart. One can first search for a critical subgraph of size at most (c|H|^{2}); if none is found, the extension exists and can be constructed by standard list‑coloring techniques (e.g., Thomassen’s linear‑time algorithm for 5‑list‑coloring planar graphs). The bound on the size of the obstruction makes this search feasible even for relatively large graphs, because the search space is confined to a region whose size is quadratic in the (typically small) pre‑colored set.

Finally, the paper discusses extensions and open problems. The authors conjecture that similar quadratic bounds should hold for higher list sizes (e.g., 6‑list‑coloring) and for other surface families such as the torus. They also suggest investigating whether the distance requirement can be lowered or replaced by structural constraints on the pre‑colored set (e.g., requiring it to form a tree). The techniques introduced—particularly the combination of discharging, reducible configuration analysis, and distance‑based weight reduction—provide a versatile toolkit for tackling these broader questions.

In summary, the paper delivers a decisive affirmative answer to Albertson’s question, establishes a concrete quadratic bound on critical subgraph size relative to the pre‑colored region, and opens a pathway toward efficient algorithms for list‑coloring planar graphs with distant pre‑colored vertices.