Abstract
A k-proper connected 2-coloring for a graph is an edge bipartition which ensures the existence of at least k vertex disjoint simple alternating paths (i.e., paths where no two adjacent edges belong to the same partition) between all pairs of vertices. In this work, for every \(k \in \mathbb {N}_{>0}\), we show that exactly counting such colorings is \(\#P\)-hard under many-one counting reductions, as well as \(\#P\)-complete under many-one counting reductions for \(k=1\). Furthermore, for every \(k \in \mathbb {N}_{>0}\), we rule out the existence of a \(2^{o\left( \frac{n}{k^2}\right) }\) time algorithm for finding a k-proper connected 2-coloring of an order n graph under the ETH, or for exactly counting such colorings assuming the moderated Counting Exponential Time Hypothesis (#ETH) of (Dell et al.; ACM Trans. Algorithms10(4); 2014). Finally, assuming the Exponential Time Hypothesis (ETH), and as a consequence of a recent result of (Dell & Lapinskas; ACM Trans. Comput. Theory13(2); 2021), for every \(k \in \mathbb {N}_{>0}\) and every \(\epsilon > 0\), we are able to rule out the existence of a \(2^{o\left( \frac{n}{k^2}\right) }/\epsilon ^2\) time algorithm for approximating the number of k-proper connected 2-colorings of an order n graph within a multiplicative factor of \(1 + \epsilon \).
