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This article is cited in 1 scientific paper (total in 1 paper)
Discrete mathematics and mathematical cybernetics
Path partitioning planar graphs of girth 4 without adjacent short cycles
A. N. Glebov, D. Zh. Zambalayeva Sobolev Institute of Mathematics,
pr. Koptyuga, 4,
630090, Novosibirsk, Russia
Abstract:
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, Lovász and Mihók posed the following Path Partition Conjecture: every graph is $\tau$-partitionable. In 2007, we 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.
Keywords:
graph, planar graph, girth, triangle-free graph, path partition, $\tau$-partitionable graph, path partition conjecture.
Received November 30, 2017, published September 21, 2018
Citation:
A. N. Glebov, D. Zh. Zambalayeva, “Path partitioning planar graphs of girth 4 without adjacent short cycles”, Sib. Èlektron. Mat. Izv., 15 (2018), 1040–1047
Linking options:
https://www.mathnet.ru/eng/semr978 https://www.mathnet.ru/eng/semr/v15/p1040
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