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 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 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.

 Funding Agency Grant Number Russian Science Foundation 16-11-10054

DOI: https://doi.org/10.17377/semi.2018.15.087

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Bibliographic databases:

Document Type: Article
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 1040--1047 \mathnet{http://mi.mathnet.ru/semr978} \crossref{https://doi.org/10.17377/semi.2018.15.087} `