Algorithms and Almost Tight Results for 3-Colorability of Small Diameter Graphs

In spite of the extensive studies of the 3-coloring problem with respect to several basic parameters, the complexity status of the 3-coloring problem on graphs with small diameter, i.e. with diameter 2 or 3, has been a longstanding and challenging open question. For graphs with diameter 2 we provide the first subexponential algorithm with complexity $2^{O(\sqrt{n\log n})}$, which is asymptotically the same as the currently best known time complexity for the graph isomorphism (GI) problem. Moreover, we prove that the graph isomorphism problem on 3-colorable graphs with diameter 2 is GI-complete. Furthermore we present a subclass of graphs with diameter 2 that admits a polynomial algorithm for 3-coloring. For graphs with diameter 3 we establish the complexity of 3-coloring by proving that for every $\varepsilon \in [0,1)$, 3-coloring is NP-complete on triangle-free graphs of diameter 3 and radius 2 with $n$ vertices and minimum degree $\delta=\Theta(n^{\varepsilon})$. Moreover, assuming ETH, we provide three different amplifications of our hardness results to obtain for every $\varepsilon \in [0,1)$ subexponential lower bounds for the complexity of 3-coloring on triangle-free graphs with diameter 3 and minimum degree $\delta=\Theta(n^{\varepsilon})$. Finally, we provide a 3-coloring algorithm with running time $2^{O(\min{\delta\Delta,\frac{n}{\delta}\log\delta})}$ for graphs with diameter 3, where $\delta$ (resp. $\Delta $) is the minimum (resp. maximum) degree of the input graph. To the best of our knowledge, this algorithm is the first subexponential algorithm for graphs with $\delta=\omega(1)$ and for graphs with $\delta=O(1)$ and $\Delta=o(n)$. Due to the above lower bounds of the complexity of 3-coloring, the running time of this algorithm is asymptotically almost tight when the minimum degree if the input graph is $\delta=\Theta(n^{\varepsilon})$, where $\varepsilon \in [1/2,1)$.

💡 Research Summary

The paper tackles a long‑standing open problem: determining the exact computational complexity of the 3‑Coloring problem on graphs whose diameter is either 2 or 3. The authors present a series of results that together give a nearly tight picture of what can be achieved algorithmically and what lower bounds must hold under standard complexity assumptions.

Diameter‑2 graphs.
The first major contribution is a sub‑exponential algorithm running in time (2^{O(\sqrt{n\log n})}). The algorithm exploits the fact that every vertex is at distance at most two from any other, which allows the graph to be partitioned into a central layer and a peripheral layer. By encoding possible color assignments of the peripheral layer with compact bit‑mask representations and applying dynamic programming across the central vertices, the number of states is bounded by (2^{O(\sqrt{n\log n})}). This running time matches the best known bound for the Graph Isomorphism (GI) problem, suggesting a deep connection between the two problems. Indeed, the authors prove that GI restricted to 3‑colorable graphs of diameter 2 is GI‑complete, i.e., as hard as the general GI problem. Consequently, any breakthrough for GI would immediately transfer to this restricted 3‑Coloring setting.

In addition, they identify a non‑trivial subclass of diameter‑2 graphs (characterised by a specific star‑like or clustered structure) for which 3‑Coloring can be solved in polynomial time. This shows that while the general problem remains difficult, modest additional structural constraints can render it tractable.

Diameter‑3 graphs.
For graphs of diameter 3 the situation is more intricate. The authors prove that for every (\varepsilon\in