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Zapiski Nauchnykh Seminarov POMI, 2016, Volume 450, Pages 5–13 (Mi znsl6333)  

On the connection between the chromatic number of a graph and the number of cycles, covering a vertex or an edge

S. L. Berlov, K. I. Tyschuk

Physical and Mathematical Lyceum 239, St. Petersburg, Russia
References:
Abstract: We prove several tight bounds on the chromatic number of a graph in terms of the minimal number of simple cycles, covering a vertex or an edge of this graph. Namely, we prove that $\chi(G)\leq k$ in the following two cases: any edge of $G$ is covered by less than $[e(k-1)!-1]$ simple cycles or any vertex of $G$ is covered by less than $[\frac{ek!}2-\frac{k+1}2]$ simple cycles. Tight bounds on the number of simple cycles covering an edge or a vertex of a $k$-critical graph are also proved.
Key words and phrases: proper coloring, chromatic number.
Funding agency Grant number
Russian Foundation for Basic Research 13-01-00563
Received: 11.10.2016
English version:
Journal of Mathematical Sciences (New York), 2018, Volume 232, Issue 1, Pages 1–5
DOI: https://doi.org/10.1007/s10958-018-3853-6
Bibliographic databases:
Document Type: Article
UDC: 519.174.7
Language: Russian
Citation: S. L. Berlov, K. I. Tyschuk, “On the connection between the chromatic number of a graph and the number of cycles, covering a vertex or an edge”, Combinatorics and graph theory. Part VIII, Zap. Nauchn. Sem. POMI, 450, POMI, St. Petersburg, 2016, 5–13; J. Math. Sci. (N. Y.), 232:1 (2018), 1–5
Citation in format AMSBIB
\Bibitem{BerTys16}
\by S.~L.~Berlov, K.~I.~Tyschuk
\paper On the connection between the chromatic number of a~graph and the number of cycles, covering a~vertex or an edge
\inbook Combinatorics and graph theory. Part~VIII
\serial Zap. Nauchn. Sem. POMI
\yr 2016
\vol 450
\pages 5--13
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl6333}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3582949}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2018
\vol 232
\issue 1
\pages 1--5
\crossref{https://doi.org/10.1007/s10958-018-3853-6}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85047428749}
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  • https://www.mathnet.ru/eng/znsl/v450/p5
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