  In a $(k,2)$-Constraint Satisfaction Problem we are given a set of arbitrary
constraints on pairs of $k$-ary variables, and are asked to find an assignment
of values to these variables such that all constraints are satisfied. The
$(k,2)$-CSP problem generalizes problems like $k$-coloring and
$k$-list-coloring. In the Unique $(k,2)$-CSP problem, we add the assumption
that the input set of constraints has at most one satisfying assignment.
  Beigel and Eppstein gave an algorithm for $(k,2)$-CSP running in time
$O\left(\left(0.4518k\right)^n\right)$ for $k>3$ and $O\left(1.356^n\right)$
for $k=3$, where $n$ is the number of variables. Feder and Motwani improved
upon the Beigel-Eppstein algorithm for $k\geq 11$. Hertli, Hurbain, Millius,
Moser, Scheder and Szedl{\'a}k improved these bounds for Unique $(k,2)$-CSP for
every $k\geq 5$.
  We improve the result of Hertli et al. and obtain better bounds for
Unique~$(k,2)$-CSP for~$k\geq 5$. In particular, we improve the running time of
Unique~$(5,2)$-CSP from~$O\left(2.254^n\right)$ to~$O\left(2.232^n\right)$ and
Unique~$(6,2)$-CSP from~$O\left(2.652^n\right)$ to~$O\left(2.641^n\right)$.
