an:06872659
Zbl 1387.05055
Glebov, Aleksey N.
Splitting a planar graph of girth 5 into two forests with trees of small diameter
EN
Discrete Math. 341, No. 7, 2058-2067 (2018).
0012-365X
2018
j
05C10 05C70 05C15
planar graph; path partition; \(P_k\)-free colouring; list colouring; forest
Summary: \textit{P. Mihók} [in: Graphs, hypergraphs and matroids. Proceedings of the fifth regional scientific session of mathematicians held in Żagań, Poland, June 1985. Section: Graph theory. Zielona Góra: Higher College of Engineering. 49--58 (1985; Zbl 0623.05043)] and recently \textit{M. Axenovich} et al. [J. Graph Theory 85, No. 3, 601--618 (2017; Zbl 1367.05044)] asked about the minimum integer \(g^\ast > 3\) such that every planar graph with girth at least \(g^\ast\) admits a 2-colouring of its vertices where the length of every monochromatic path is bounded from above by a constant. By results of \textit{A. N. Glebov} and \textit{D. Zh. Zambalaeva} [Sib. Èlektron. Mat. Izv. 4, 450--459 (2007; Zbl 1132.05315)] and of M. Axenovich et al. [loc. cit.], it follows that \(5 \leq g^\ast \leq 6\). In this paper we establish that \(g^\ast = 5\). Moreover, we prove that every planar graph of girth at least 5 admits a 2-colouring of its vertices such that every monochromatic component is a tree of diameter at most 6. We also present the list version of our result.
0623.05043; 1367.05044; 1132.05315