an:06758793
Zbl 1367.05044
Axenovich, Maria; Ueckerdt, Torsten; Weiner, Pascal
Splitting planar graphs of girth 6 into two linear forests with short paths
EN
J. Graph Theory 85, No. 3, 601-618 (2017).
00368181
2017
j
05C10 05C12 05C38 05C70 05C15
planar graph decomposition; defective coloring; fragmented coloring; path forests; 2-coloring
Summary: Recently, \textit{O. V. Borodin} et al. [Discrete Math. 313, No. 22, 2638--2649 (2013; Zbl 1281.05060)] showed that the vertices of each planar graph of girth at least 7 can be 2-colored so that each color class induces a subgraph of a matching. We prove that any planar graph of girth at least 6 admits a vertex coloring in two colors such that each monochromatic component is a path of length at most 14. Moreover, we show a list version of this result. On the other hand, for each positive integer \(t\geq 3\), we construct a planar graph of girth 4 such that in any coloring of vertices in two colors there is a monochromatic path of length at least \(t\). It remains open whether each planar graph of girth 5 admits a 2-coloring with no long monochromatic paths.
Zbl 1281.05060