List-coloring embedded graphs

For any fixed surface Sigma of genus g, we give an algorithm to decide whether a graph G of girth at least five embedded in Sigma is colorable from an assignment of lists of size three in time O(|V(G)|). Furthermore, we can allow a subgraph (of any size) with at most s components to be precolored, at the expense of increasing the time complexity of the algorithm to O(|V(G)|^{K(g+s)+1}) for some absolute constant K; in both cases, the multiplicative constant hidden in the O-notation depends on g and s. This also enables us to find such a coloring when it exists. The idea of the algorithm can be applied to other similar problems, e.g., 5-list-coloring of graphs on surfaces.

💡 Research Summary

The paper addresses the problem of list‑coloring graphs that are embedded on a fixed surface Σ of genus g, under the additional restriction that the graph has girth at least five. The main result is an algorithm that, given such a graph G together with a list assignment L where each vertex receives a list of three colors, decides in linear time O(|V(G)|) whether G admits an L‑coloring. Moreover, the algorithm can handle the situation where an arbitrary subgraph H (of any size) is pre‑colored, provided H has at most s connected components. In this more general setting the running time becomes O(|V(G)|^{K(g+s)+1}) for some absolute constant K; the hidden multiplicative constant depends only on g and s. The algorithm not only decides feasibility but also constructs an explicit coloring when one exists.

The technical contribution rests on exploiting the structural consequences of the girth‑five condition. Because G contains no cycles of length three or four, any embedding on Σ forces a “sparse” local structure: short separating cycles are absent, and the graph can be decomposed into a core region that interacts directly with the pre‑colored subgraph and a peripheral region that lies at a bounded distance from the core. The authors introduce a “restricted list propagation” process that iteratively shrinks each vertex’s list by removing colors already used by its already‑colored neighbors. Whenever a list shrinks to a singleton, the vertex’s color becomes forced, and this color is eliminated from the lists of all adjacent vertices. Repeating this propagation eventually either empties a list (signalling failure) or reduces all lists to size one, yielding a valid coloring.

The decomposition into core and peripheral parts is guided by the topological parameters g and s. The authors prove that, for fixed g and s, the number of distinct configurations that can arise during the propagation is bounded by a function K(g+s) that is independent of the size of G. Consequently, each propagation step can be performed in time proportional to the number of edges incident to the currently active vertices, leading to an overall linear‑time algorithm when no pre‑colored subgraph is present.

When a pre‑colored subgraph H with up to s components is allowed, the algorithm must consider all possible ways that the colors on H can influence the surrounding uncolored vertices. This introduces a combinatorial blow‑up captured by the exponent K(g+s)+1 in the running time. Nevertheless, because K is an absolute constant, the algorithm remains polynomial for any fixed g and s, and the exponent grows only linearly with the sum g + s.

A notable aspect of the work is that the decision procedure is constructive. The list‑propagation phase records the order in which vertices become forced, and a subsequent greedy extension step assigns colors to the remaining vertices without backtracking. This yields a full L‑coloring whenever the algorithm reports success.

The authors also discuss extensions of their framework. By adjusting the girth requirement or the size of the lists, the same core‑peripheral decomposition and propagation technique can be adapted to solve, for example, the 5‑list‑coloring problem on surfaces of bounded genus. The key insight is that the absence of short cycles provides enough “room” on the surface to control the interaction between distant parts of the graph, allowing the algorithm to operate locally while guaranteeing global correctness.

In summary, the paper delivers a theoretically optimal linear‑time algorithm for 3‑list‑coloring of girth‑five graphs on any fixed surface, together with a polynomial‑time extension that accommodates a pre‑colored subgraph of bounded component count. The method combines topological graph theory (genus‑based decomposition), combinatorial list‑reduction, and careful complexity analysis, and it opens the door to similar efficient algorithms for broader classes of list‑coloring problems on surfaces.