In 1976, Steinberg conjectured that planar graphs without 4-cycles and
5-cycles are 3-colorable. This conjecture attracted numerous researchers for
about 40 years until it was disproved by Cohen-Addad et al. in 2017. How-
ever, coloring planar graphs with restrictions on cycle lengths is still an active
area of research, and the interest in this particular graph class remains.
Recently, Cho, Choi, Park (2021) showed that for a planar graph G
without 4-cycles and 5-cycles, V (G) is partitioned into two sets A and B
such that G[A] and G[B] are forests with maximum degree three and four,
respectively. In this thesis, we show that for a planar graph G without 4-
cycles and 5-cycles, V (G) is partitioned into two sets A and B such that
G[A] is a linear forest and G[B] has maximum degree at most 8.
