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Zapiski Nauchnykh Seminarov LOMI, 1979, Volume 88, Pages 209–217 (Mi znsl3115)  

Representation of proof s' by coloured graphs and Hadwiger hypothesis

P. Yu. Suvorov
Abstract: Coloured graph is a directed graph with some vertices and ares coloured in some colours.
If $A$ is a set of $k$-vertex coloured graphs and $G$ is a coloured graph then $G$ is said to be $A$-well-coloured if all its $k$-vertex subgraphs belong to $A$.
We describe a construction of a finite set $B_p$ of 3-vertex graphs with ares coloured in some colours and a finite set $C_p$ of 2-vertex graphs with ares and vertices coloured in some colours for a given formula $P$ of the first order predicate calcules such that $\vdash P$ if and only if there exists $B_p$-well-(arc)coloured graph $G$ which does not permit $C_p$-well-colouring of vertices.
Hadwiger hypothesis (HH): if no subgraph of lopp-free graph $G$ is contracted to $n$-vertex complete graph, then vertices of $G$ can be coloured in $(n-1)$ colours in such a way that neighbouring vertices are coloured in distinct colours.
We construct a formula $X$ of the first order predicate calcules such that HH is equivalent to $\rceil\vdash X$. Thus HH is reduced to the statement.
If all 3-vertex subgraphs of arc-coloured graph $G$ belong to $B_X$ then the vertices of $G$ can be $C_X$-well-coloured.
Here $B_X$ and $C_X$ are concrete finite sets of 3-vertex and 2-vertex coloured graphs.
English version:
Journal of Soviet Mathematics, 1982, Volume 20, Issue 4, Pages 2376–2381
DOI: https://doi.org/10.1007/BF01629450
Bibliographic databases:
UDC: 510.66+519.174
Language: Russian
Citation: P. Yu. Suvorov, “Representation of proof s' by coloured graphs and Hadwiger hypothesis”, Studies in constructive mathematics and mathematical logic. Part VIII, Zap. Nauchn. Sem. LOMI, 88, "Nauka", Leningrad. Otdel., Leningrad, 1979, 209–217; J. Soviet Math., 20:4 (1982), 2376–2381
Citation in format AMSBIB
\Bibitem{Suv79}
\by P.~Yu.~Suvorov
\paper Representation of proof s' by coloured graphs and Hadwiger hypothesis
\inbook Studies in constructive mathematics and mathematical logic. Part~VIII
\serial Zap. Nauchn. Sem. LOMI
\yr 1979
\vol 88
\pages 209--217
\publ "Nauka", Leningrad. Otdel.
\publaddr Leningrad
\mathnet{http://mi.mathnet.ru/znsl3115}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=556232}
\zmath{https://zbmath.org/?q=an:0429.05039|0494.05022}
\transl
\jour J. Soviet Math.
\yr 1982
\vol 20
\issue 4
\pages 2376--2381
\crossref{https://doi.org/10.1007/BF01629450}
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