  We present two new quantum algorithms that either find a triangle (a copy of
$K_{3}$) in an undirected graph $G$ on $n$ nodes, or reject if $G$ is triangle
free. The first algorithm uses combinatorial ideas with Grover Search and makes
$\tilde{O}(n^{10/7})$ queries. The second algorithm uses $\tilde{O}(n^{13/10})$
queries, and it is based on a design concept of Ambainis~\cite{amb04} that
incorporates the benefits of quantum walks into Grover search~\cite{gro96}. The
first algorithm uses only $O(\log n)$ qubits in its quantum subroutines,
whereas the second one uses O(n) qubits. The Triangle Problem was first treated
in~\cite{bdhhmsw01}, where an algorithm with $O(n+\sqrt{nm})$ query complexity
was presented, where $m$ is the number of edges of $G$.
