In this paper, a new type of edge color-ing of graphs together with an algorithm for such an edge coloring is presented to construct some column-weight three low-density parity-check (LDPC) codes whose Tanner graphs are free of 4-cycles. This kind of edge coloring is applied on some well-known classes of graphs such as complete graphs and complete bipar-tite graphs to generate some column-weight 3 LDPC codes having flexibility in terms of code length and rate. Interestingly, the constructed (3, k)-regular codes with regularities k = 4, 5, . . . , 22 have lengths n =12, 20, 26, 35, 48, 57, 70, 88, 104, 117, 140, 155, 176, 204, 228, 247, 280, 301, 330, having minimum block length compared to the best known similar codes in the literature. In addition to linear complexity of generat-ing such parity-check matrices, they can be considered as the base matrices of some quasi-cyclic (QC) LDPC codes with maximum achievable girth 18, which in-herit the low-complexity encoder implementations of QC-LDPC codes. Simulation results show that the QC-LDPC codes with large girth lifted from the con-structed base matrices have good performances and outperform random codes, progressive edge growth LDPC codes, somefinitefields and group rings based QC-LDPC codes and also have a close competition to the standard IEEE 802. 16e (WiMAX) code.