Partitioning planar graphs without 4-cycles and 5-cycles into bounded degree forests
Abstract
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 recently disproved by Cohen-Addad et al. (2017). However, coloring planar graphs with restrictions on cycle lengths is still an active area of research, and the interest in this particular graph class remains. Let G be a planar graph without 4-cycles and 5-cycles. For integers d1 and d2 satisfying d1+d2≥8 and d2≥ d1≥ 2, it is known that V(G) can be partitioned into two sets V1 and V2, where each Vi induces a graph with maximum degree at most di. Since Steinberg's Conjecture is false, a partition of V(G) into two sets, where one induces an empty graph and the other induces a forest is not guaranteed. Our main theorem is at the intersection of the two aforementioned research directions. We prove that V(G) can be partitioned into two sets V1 and V2, where V1 induces a forest with maximum degree at most 3 and V2 induces a forest with maximum degree at most 4; this is both a relaxation of Steinberg's conjecture and a strengthening of results by Sittitrai and Nakprasit (2019) in a much stronger form.