
Sib. Èlektron. Mat. Izv., 2018, Volume 15, Pages 1040–1047
(Mi semr978)




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 trianglefree 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, trianglefree graph, path partition, $\tau$partitionable graph, path partition conjecture.
DOI:
https://doi.org/10.17377/semi.2018.15.087
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UDC:
519.172.2, 519.174
MSC: 05C10, 05C15, 05C70 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
Citation in format AMSBIB
\Bibitem{GleZam18}
\by A.~N.~Glebov, D.~Zh.~Zambalayeva
\paper Path partitioning planar graphs of girth 4 without adjacent short cycles
\jour Sib. \`Elektron. Mat. Izv.
\yr 2018
\vol 15
\pages 10401047
\mathnet{http://mi.mathnet.ru/semr978}
\crossref{https://doi.org/10.17377/semi.2018.15.087}
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