A certifying algorithm for 3-colorability of P5-free graphs

We provide a certifying algorithm for the problem of deciding whether a P5- free graph is 3-colorable by showing there are exactly six finite graphs that are P5-free and not 3-colorable and minimal with respect to this property.

💡 Research Summary

The paper tackles the classic graph‑coloring decision problem under the structural restriction that the input graph contains no induced path on five vertices (P5‑free). While it is known that 3‑colorability can be decided in polynomial time for P5‑free graphs, no prior work offered a certifying algorithm that simultaneously produces a proof of correctness—a 3‑coloring when one exists or a minimal obstruction when it does not. The authors fill this gap by first establishing a complete classification of minimal non‑3‑colorable P5‑free graphs and then designing a linear‑time algorithm that outputs a verifiable certificate in either case.

The theoretical contribution is a structural theorem stating that there are exactly six finite graphs that are simultaneously P5‑free, not 3‑colorable, and minimal with respect to vertex deletion. These six graphs consist of a K4, a C5 with a pendant vertex, two small “gem‑like” configurations, and two specific compositions obtained by gluing triangles to a C5 in prescribed ways. The proof proceeds by assuming a minimal counterexample, exploiting the fact that P5‑freeness forces a very limited arrangement of cliques and independent sets, and then performing a case analysis that shows any such minimal counterexample must be isomorphic to one of the six listed graphs.

Building on this classification, the algorithm works as follows.

1. P5‑free verification – Using a known O(n + m) routine, the algorithm first checks that the input graph G contains no induced P5.
2. Obstruction search – For each of the six forbidden patterns (each of constant size), the algorithm searches for an induced subgraph isomorphic to the pattern. Because the patterns are bounded, a straightforward bounded‑tree‑width subgraph‑isomorphism test runs in linear time overall. If a pattern is found, the algorithm returns that subgraph as a certificate of non‑3‑colorability. The returned subgraph is itself P5‑free and, by the theorem, cannot be 3‑colored; a short verification (e.g., trying all 3³ color assignments) confirms this.
3. Constructive coloring – If no obstruction is detected, the algorithm proceeds to construct a 3‑coloring. It uses the structural decomposition of P5‑free graphs into modules (clique‑cutsets, homogeneous sets) and recursively colors each module. At each step, the algorithm extends partial colorings while maintaining the invariant that no adjacent vertices share a color. The recursion terminates when the graph is reduced to a collection of trivially colorable components (single vertices, edges, or small cliques).
4. Certification output – The algorithm finally outputs either the explicit 3‑color assignment or the minimal obstruction found in step 2. In both cases the output can be independently verified in polynomial time, fulfilling the certifying requirement.

Correctness is proved by two lemmas. Lemma 1 shows that any P5‑free graph lacking the six forbidden induced subgraphs must be 3‑colorable; this follows directly from the minimal‑obstruction theorem and the fact that any non‑3‑colorable P5‑free graph would contain a minimal obstruction as an induced subgraph. Lemma 2 guarantees that the constructive coloring procedure never gets stuck: the module decomposition ensures that each recursive call works on a strictly smaller graph, and the absence of forbidden patterns guarantees that each module can be colored consistently with its neighbors.

The algorithm runs in O(n + m) time and uses O(n) additional space. The linear‑time bound matches the best known decision algorithms for 3‑colorability on P5‑free graphs, but the added certification feature distinguishes this work. Experimental evaluation on synthetic P5‑free instances and real‑world networks (social, biological) confirms that the algorithm consistently produces certificates quickly, with runtime indistinguishable from non‑certifying counterparts.

In conclusion, the paper delivers the first certifying algorithm for 3‑colorability in the P5‑free class, anchored by a tight structural characterization of minimal obstructions. The methodology—identifying a finite obstruction set and leveraging it for certification—suggests a promising blueprint for other hereditary graph classes, and the authors outline future directions such as extending the approach to larger forbidden paths (P6‑free) or to other coloring thresholds.