  Given a 3-colorable graph $X$, the 3-coloring complex $B(X)$ is the graph
whose vertices are all the independent sets which occur as color classes in
some 3-coloring of $X$. Two color classes $C,D \in V(B(X))$ are joined by an
edge if $C$ and $D$ appear together in a 3-coloring of $X$. The graph $B(X)$ is
3-colorable. Graphs for which $B(B(X))$ is isomorphic to $X$ are termed
reflexive graphs. In this paper, we consider 3-edge-colorings of cubic graphs
for which we allow half-edges. Then we consider the 3-coloring complexes of
their line graphs. The main result of the paper is a surprising outcome that
the line graph of any connected cubic triangle-free outerplanar graph is
reflexive. We also exhibit some other interesting classes of reflexive line
graphs.
