  A path (cycle) is properly-colored if consecutive edges are of distinct
colors. In 1997, Bang-Jensen and Gutin conjectured a necessary and sufficient
condition for the existence of a Hamilton path in an edge-colored complete
graph. This conjecture, confirmed by Feng, Giesen, Guo, Gutin, Jensen and
Rafley in 2006, was laterly playing an important role in Lo's asymptotical
proof of Bollob\'as-Erd\H{o}s' conjecture on properly-colored Hamilton cycles.
In 1997, Yeo obtained a structural characterization of edge-colored graphs that
containing no properly colored cycles. This result is a fundamental tool in the
study of edge-colored graphs. In this paper, we first give a much shorter proof
of the Bang-Jensen-Gutin Conjecture by two novel absorbing lemmas. We also
prove a new sufficient condition for the existence of a properly-colored cycle
and then deduce Yeo's theorem from this result and a closure concept in
edge-colored graphs.
