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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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