Three-coloring triangle-free graphs on surfaces I. Extending a coloring to a disk with one triangle

Let G be a plane graph with exactly one triangle T and all other cycles of length at least 5, and let C be a facial cycle of G of length at most six. We prove that a 3-coloring of C does not extend to a 3-coloring of G if and only if C has length exactly six and there is a color x such that either G has an edge joining two vertices of C colored x, or T is disjoint from C and every vertex of T is adjacent to a vertex of C colored x. This is a lemma to be used in a future paper of this series.

💡 Research Summary

The paper investigates a very specific extension problem for 3‑colorings of planar graphs that contain exactly one triangle. Let G be a plane graph in which the unique triangle is denoted by T and every other cycle has length at least five. Let C be a facial cycle of G whose length does not exceed six, and suppose a proper 3‑coloring φ of C is given. The authors determine precisely when φ cannot be extended to a proper 3‑coloring of the whole graph G. Their main theorem states that non‑extendability occurs if and only if two conditions are simultaneously satisfied: (i) C has length exactly six, and (ii) there exists a color x such that either (a) G contains an edge whose both endpoints lie on C and are colored x by φ, or (b) the triangle T is disjoint from C and each vertex of T is adjacent to a vertex of C that receives color x. In all other configurations—when C is shorter than six, when no edge of C joins two vertices of the same color, or when T either shares a vertex with C or is not completely dominated by a single color on C—the coloring φ can always be extended to G.

The proof proceeds by contradiction using a minimal counterexample. Assuming a smallest pair (G, φ) for which φ does not extend, the authors first eliminate the possibility that |C| ≤5, showing that standard recoloring techniques (Kempe chain arguments, boundary swaps) always succeed in that case. Hence any counterexample must have |C| = 6. They then analyze the structure of G around C. If an edge of C connects two vertices colored x, the obstruction is immediate: the edge forces a monochromatic adjacency, violating proper coloring. If no such edge exists, the focus shifts to the relationship between T and C. When T shares a vertex with C, a recoloring of the shared vertex resolves any conflict, so T must be completely separate from C. The authors then prove that if every vertex of T is adjacent to a vertex of C colored x, the triangle is forced to inherit color x on all three of its vertices, making a proper 3‑coloring impossible. Conversely, if at least one vertex of T lacks a neighbor of color x on C, a Kempe‑chain recoloring can be performed to free a color for that vertex, allowing the extension.

Throughout the argument the authors exploit the minimality of the counterexample to delete irrelevant edges or contract degree‑2 vertices without destroying the essential properties (single triangle, cycle‑length condition). This reduction process eventually yields a configuration that contradicts the assumed minimality, thereby confirming that the two described scenarios are the only obstacles to extension.

The result serves as a crucial lemma for the authors’ forthcoming work, where they consider more complex embeddings of graphs on surfaces and disks that may contain additional triangles or longer cycles. By establishing a clean, combinatorial criterion for the simplest case—one triangle and a short facial cycle—the paper provides a solid foundation for inductive arguments and discharging procedures in the broader series. The theorem also enriches the landscape of planar graph coloring by pinpointing the exact structural reason why a seemingly benign boundary coloring might fail to propagate inward, a nuance that is invisible in the classic Grötzsch theorem (triangle‑free case) but becomes decisive once a single triangle is permitted.