An {\em acyclic edge coloring} of a graph $G$ is a proper edge coloring such
that the subgraph induced by any two color classes is a linear forest (an
acyclic graph with maximum degree at most two). The {\em acyclic chromatic
index} $\chiup_{a}'(G)$ of a graph $G$ is the least number of colors needed in
an acyclic edge coloring of $G$. Fiam\v{c}\'{i}k (1978) conjectured that
$\chiup_{a}'(G) \leq \Delta(G) + 2$, where $\Delta(G)$ is the maximum degree of
$G$. This conjecture is well known as Acyclic Edge Coloring Conjecture (AECC).
A graph $G$ with maximum degree at most $\kappa$ is {\em
$\kappa$-deletion-minimal} if $\chiup_{a}'(G) > \kappa$ and $\chiup_{a}'(H)
\leq \kappa$ for every proper subgraph $H$ of $G$. The purpose of this paper is
to provide many structural lemmas on $\kappa$-deletion-minimal graphs. By using
the structural lemmas, we firstly prove that AECC is true for the graphs with
maximum average degree less than four (\autoref{NMAD4}). We secondly prove that
AECC is true for the planar graphs without triangles adjacent to cycles of
length at most four, with an additional condition that every $5$-cycle has at
most three edges contained in triangles (\autoref{NoAdjacent}), from which we
can conclude some known results as corollaries. We thirdly prove that every
planar graph $G$ without intersecting triangles satisfies $\chiup_{a}'(G) \leq
\Delta(G) + 3$ (\autoref{NoIntersect}). Finally, we consider one extreme case
and prove it: if $G$ is a graph with $\Delta(G) \geq 3$ and all the
$3^{+}$-vertices are independent, then $\chiup_{a}'(G) = \Delta(G)$. We hope
the structural lemmas will shed some light on the acyclic edge coloring
problems.Comment: 19 page