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

Abstract

The 3-coloring problem is well known to be NP-complete. It is also well known that it remains NP-complete when the input is restricted to graphs with diameter 4. Moreover, assuming the Exponential Time Hypothesis (ETH), 3-coloring can not be solved in time 2^o(n) on graphs with n vertices and diameter at most 4. In spite of the extensive studies of the 3-coloring problem with respect to several basic parameters, the complexity status of this problem on graphs with small diameter, i.e. with diameter at most 2, or at most 3, has been a longstanding and challenging open question. In this paper we investigate graphs with small diameter. For graphs with diameter at most 2, we provide the first subexponential algorithm for 3-coloring, with complexity $2^{O(\sqrt{n\log n})}$. Furthermore we present a subclass of graphs with diameter 2 that admits a polynomial algorithm for 3-coloring. For graphs with diameter at most 3, we establish the complexity of 3-coloring, even for the case of triangle-free graphs. Namely we prove that for every ${\varepsilon \in \lbrack 0,1)}$, 3-coloring is NP-complete on triangle-free graphs of diameter 3 and radius 2 with n vertices and minimum degree δ = Θ(n ^ε). Moreover, assuming ETH, we use three different amplification techniques of our hardness results, in order to obtain for every ${\varepsilon \in \lbrack 0,1)}$ subexponential asymptotic lower bounds for the complexity of 3-coloring on triangle-free graphs with diameter 3 and minimum degree δ = Θ(n ^ε). Finally, we provide a 3-coloring algorithm with running time ${2^{O(\min \{\delta \Delta ,\ \frac{n}{\delta }\log \delta \})}}$ for arbitrary graphs with diameter 3, where n is the number of vertices and δ (resp. Δ) 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 δ = ω(1) and for graphs with δ = O(1) and Δ = 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 of the input graph is δ = Θ(n ^ε), where $\varepsilon \in \lbrack \frac{1}{2},1)$.