an:07299431
Zbl 1455.05059
Cho, Eun-Kyung; Choi, Ilkyoo; Park, Boram
Partitioning planar graphs without 4-cycles and 5-cycles into bounded degree forests
EN
Discrete Math. 344, No. 1, Article ID 112172, 9 p. (2021).
00457331
2021
j
05C70 05C10
Summary: 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 \textit{V. Cohen-Addad} et al. [J. Comb. Theory, Ser. B 122, 452--456 (2017; Zbl 1350.05018)]. 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 \(d_1\) and \(d_2\) satisfying \(d_1 + d_2 \geq 8\) and \(d_2 \geq d_1 \geq 2\), it is known that \(V(G)\) can be partitioned into two sets \(V_1\) and \(V_2\), where each \(V_i\) induces a graph with maximum degree at most \(d_i\). 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 \(V_1\) and \(V_2\), where \(V_1\) induces a forest with maximum degree at most 3 and \(V_2\) induces a forest with maximum degree at most 4; this is both a relaxation of Steinberg's conjecture and a strengthening of results by \textit{P. Sittitrai} and \textit{K. Nakprasit} [Discrete Math. 341, No. 8, 2142--2150 (2018; Zbl 1388.05072)] in a much stronger form.
Zbl 0791.05044; Zbl 1350.05018; Zbl 1388.05072