Coloring Problems on Bipartite Graphs of Small Diameter

💡 Research Summary

The paper investigates a suite of coloring and homomorphism problems on bipartite graphs whose diameter is bounded by a small constant. The authors focus on three canonical coloring formulations: k‑List Coloring, List k‑Coloring, and k‑Precoloring Extension (k‑PreExt). While it is classical that k‑List Coloring is NP‑complete for any fixed k ≥ 3, the impact of a diameter restriction on the complexity has been largely unexplored.

The main contributions are as follows. First, the authors prove that for bipartite graphs of diameter at most four, the problems 3‑PreExt, Edge‑Surjective C₆‑Homomorphism, and Surjective C₆‑Homomorphism are all NP‑complete. The NP‑completeness of Surjective C₆‑Homomorphism had been shown earlier by Vikas (2017) using a construction that adds a quadratic number of auxiliary vertices; the present work supplies a much simpler reduction that needs only a linear number of new vertices, thereby clarifying the underlying structure.

A key technical ingredient is a strengthened NP‑completeness result for the Retract‑to‑C₆ problem. The authors show that even when the bipartite instance (X ∪ Y, E) satisfies two additional constraints—(i) the Y‑side vertices dominate X, and (ii) every vertex of Y lies at distance at most two from any other vertex of Y—the retract problem remains NP‑complete. This refined hardness is then used to establish the NP‑completeness of 3‑PreExt on bipartite graphs of diameter four.

By systematically combining these reductions, the paper provides an almost complete classification of the three coloring problems with respect to the pair (k, d), where k is the number of colors and d is the maximum allowed diameter. All cases are shown to be NP‑complete except for the two open instances: List 3‑Coloring and 3‑PreExt on bipartite graphs of diameter three. Consequently, the long‑standing question of whether a graph of diameter two can be 3‑colored in polynomial time is effectively resolved for bipartite graphs, leaving only the diameter‑three scenario unresolved.

On the algorithmic side, the authors present a linear‑time algorithm for k‑PreExt on complete bipartite graphs (diameter two). This improves upon earlier XP‑time algorithms for broader graph classes such as P₅‑free or (rP₁ + P₅)‑free graphs, and demonstrates that the problem is fixed‑parameter tractable when parameterized by k in this restricted setting.

The paper also revisits the 3‑Biclique Partition problem. An earlier proof of NP‑completeness for bipartite graphs (Fleischner et al., 2009) contained a flaw; the authors identify the error and, using their Surjective C₆‑Homomorphism result, give a correct reduction that shows 3‑Biclique Partition is NP‑complete even when the bipartite complement has diameter four.

Finally, the authors study the 3‑Fall Coloring problem, where each vertex must see all three colors in its closed neighbourhood. They prove NP‑completeness on bipartite graphs of diameter four, and argue that if the same result held for diameter three, then the remaining open list‑coloring cases would be settled, thereby answering a question posed by Kratochvíl et al. (2002).

In summary, the paper delivers a comprehensive complexity landscape for several fundamental coloring and homomorphism problems on bipartite graphs with small diameter, introduces simpler reductions for surjective homomorphisms, supplies a fast algorithm for a special case of precoloring extension, and corrects the literature on biclique partition hardness. The only remaining gap—determining the status of List 3‑Coloring and 3‑PreExt on diameter‑three bipartite graphs—offers a clear direction for future research.