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Sibirskie Èlektronnye Matematicheskie Izvestiya [Siberian Electronic Mathematical Reports], 2007, Volume 4, Pages 435–439 (Mi semr165)  

This article is cited in 2 scientific papers (total in 2 papers)

Research papers

Minimax degrees of quasiplane graphs without $4$-faces

O. V. Borodina, A. O. Ivanovab, A. V. Kostochkaa, N. N. Sheikhc

a Sobolev Institute of Mathematics, Novosibirsk, Russia
b Yakutsk State University
c Department of Mathematics, University of Illinois Department of Mathematics, Urbana, USA
Full-text PDF (699 kB) Citations (2)
References:
Abstract: The $M$-degree of an edge $xy$ in a graph is the maximum of the degrees of $x$ and $y$. The minimax degree of a graph $G$ is the minimum over $M$-degrees of its edges. In order to get upper bounds on the game chromatic number, W. He et al showed that every planar graph $G$ without leaves and $4$-cycles has minimax degree at most $8$. This was improved by Borodin et al to the best possible bound $7$. Answering a question by D. West, we show that every plane graph $G$ without leaves and $4$-faces has minimax degree at most $15$. The bound is sharp. Similar results are obtained for graphs embeddable on the projective plane, torus and Klein bottle.
Received October 3, 2007, published October 16, 2007
Bibliographic databases:
Document Type: Article
UDC: 519.172.2
MSC: 05С15
Language: English
Citation: O. V. Borodin, A. O. Ivanova, A. V. Kostochka, N. N. Sheikh, “Minimax degrees of quasiplane graphs without $4$-faces”, Sib. Èlektron. Mat. Izv., 4 (2007), 435–439
Citation in format AMSBIB
\Bibitem{BorIvaKos07}
\by O.~V.~Borodin, A.~O.~Ivanova, A.~V.~Kostochka, N.~N.~Sheikh
\paper Minimax degrees of quasiplane graphs without $4$-faces
\jour Sib. \`Elektron. Mat. Izv.
\yr 2007
\vol 4
\pages 435--439
\mathnet{http://mi.mathnet.ru/semr165}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2465435}
\zmath{https://zbmath.org/?q=an:1132.05312}
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  • This publication is cited in the following 2 articles:
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