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Path partitioning planar graphs of girth 4 without adjacent short cycles. (Russian. English summary) Zbl 1398.05070
Summary: 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
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