  A famous conjecture of Caccetta and H\"{a}ggkvist (CHC) states that a
directed graph $D$ with $n$ vertices and minimum outdegree at least $r$ has a
directed cycle of length at most $\lceil \frac{n}{r}\rceil$. In 2017, Aharoni
proposed the following generalization: an edge-colored graph $G$ with $n$
vertices, $n$ color classes of size at least $r$ has a rainbow cycle of length
at most $\lceil \frac{n}{r}\rceil$. Since CHC can be seen as the case of
Aharoni's Conjecture: color classes in the color partition are monochromatic
stars centered at distinct vertices, one way to study Aharoni's Conjecture is
to structure the color classes as each color class is either a star, a triangle
or contains a matching of size 2. Guo improved the upper bound in Aharoni's
Conjecture to $O(\log n)$ in some mixed cases when the color classes are not
necessarily stars. In this paper, we extend Guo's results. Our main result is
as follows: Let $G$ an edge-colored graph on $n$ vertices and $n$ color
classes, if at least $\alpha n$ color classes are either a matching of size 2
or a triangle for $\alpha >\frac{1}{2}$, then $G$ contains a rainbow cycle of
length $O(\log n)$. We also prove that the $\log n$ bound is the right order of
magnitude.
