Path partitioning planar graphs of girth 4 without adjacent short cycles.

*(Russian. English summary)*Zbl 1398.05070Summary: A graph \(G\) is \((a, b)\)-partitionable for positive intergers \(a\), \(b\) if its vertex set can be partitioned into subsets \(V_1\), \(V_2\) such that the induced subgraph \(G[V_1]\) contains no path on \(a+1\) vertices and the induced subgraph \(G[V_2]\) contains no path on \(b + 1\) vertices. A graph \(G\) is \(\tau\)-partitionable if it is \((a, b)\)-partitionable for every pair \(a,b\) such that \(a + b\) is the number of vertices in the longest path of \(G\). In 1981, L. Lovász and P. Mihók posed in Szeged the following path partition conjecture: every graph is \(\tau\)-partitionable. The authors [Sib. Èlektron. Mat. Izv. 4, 450–459 (2007; Zbl 1132.05315)] proved the conjecture for planar graphs of girth at least 5. The aim of this paper is to improve this result by showing that every triangle-free planar graph, where cycles of length 4 are not adjacent to cycles of length 4 and 5, is \(\tau\)-partitionable.

##### MSC:

05C10 | Planar graphs; geometric and topological aspects of graph theory |

05C15 | Coloring of graphs and hypergraphs |

05C70 | Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) |

05C38 | Paths and cycles |

##### Keywords:

planar graph; girth; triangle-free graph; path partition; \(\tau\)-partitionable graph; path partition conjecture
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\textit{A. N. Glebov} and \textit{D. Z. Zambalaeva}, Sib. Èlektron. Mat. Izv. 15, 1040--1047 (2018; Zbl 1398.05070)

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