  One of the phenomena that causes high decoding failure rates is trapping
sets. Characterization of $(a,b)$ elementary trapping sets (ETSs), their
graphical properties and the lower bounds on their size in variable regular
LDPC codes with column weights 3, 4, 5 and 6, where $a$ is the size of the ETS
and $b$ is the number of degree-one check nodes, have been an interesting
subject among researchers. Although progressive-edge-growth method (PEG) to
construct LDPC codes free of an specific ETS has been proposed in the
literature, it is mostly applied to LDPC codes with column weight 3. In this
paper, we focus on constructing QC-LDPC codes with column weight 4 whose Tanner
graphs are free of small ETSs. Using coloring the edges of the variable node
(VN) graph corresponding to an ETS, we provide the sufficient conditions to
obtain QC-LDPC codes with column weight 4, girth 6 and free of $(5,b)$ ETSs,
where $b\leq4$, and $(6,b)$ ETs, where $b\leq2$. Moreover, for $(4,n)$-regular
QC-LDPC codes with girth 8, we present a method to remove $(7,4)$ ETSs from
Tanner graphs.
