main.tex

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\documentclass[
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]{ceurart}

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\usepackage{listings}
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\input definitions.tex
\begin{document}

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%% Rights management information.
%% CC-BY is default license.
\copyrightyear{2022}
\copyrightclause{Copyright for this paper by its authors.
  Use permitted under Creative Commons License Attribution 4.0
  International (CC BY 4.0).}

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\conference{ITAT’23: Information technologies -- Applications and Theory, September 22–26, 2023, Tatranské Matliare, Slovakia}

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\title{Analysis of colouring properties of proper (2, 3)-poles}


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\author[1]{Erik Řehulka}[%
email=rehulka3@uniba.sk,
]
\address[1]{Comenius University, Mlynská dolina, 842 48 Bratislava, Slovakia}

\author[1]{Jozef Rajník}[%
email=jozef.rajnik@fmph.uniba.sk,
]

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%% The abstract is a short summary of the work to be presented in the
%% article.
\begin{abstract}
  Snarks, that is $2$-connected cubic graphs admitting no $3$-edge-colouring, provide a promising family of cubic graphs with respect to many widely-open conjectures. They are often constructed by joining several building blocks which can be regarded as cubic ``graphs'' with dangling edges allowed, formally called multipoles. Colouring properties of multipoles with at most five dangling edges relevant for constructions of snarks are almost completely characterised. The remaining uncharacterised class of such multipoles are so-called proper (2,3)-poles that can be obtained by severing an edge and removing a vertex from a snark.
    
   	Therefore, in our work, we analyse the colouring properties of proper (2,3)-poles. To conduct our analysis, we explore using a computer all proper (2,3)-poles resulting from nontrivial snarks with at most 28 vertices. This encompasses a~total of 3,247 snarks and 3,476,400 proper (2,3)-poles. In our research, we provide various structures that can be utilized to expand the colourability of proper (2,3)-poles. In the core of our work, we provide theorems regarding the colouring properties of proper (2,3)-poles, specifically necessary and sufficient conditions for these properties. Additionally, we present the data and observations from the analysis.
\end{abstract}

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%% Keywords. The author(s) should pick words that accurately describe
%% the work being presented. Separate the keywords with commas.
\begin{keywords}
	snark \sep 
	multipole \sep 
	edge-colouring \sep 
	Tait colouring \sep 
	colouring set
\end{keywords}

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\input chapters/00-introduction.tex
\input chapters/10-multipolesAndSnarks.tex 
\input chapters/20-proper23poles.tex
\input chapters/40-ourWork.tex
%\input chapters/90-conclusion.tex

\bibliography{references}

\end{document}

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