Restricted frame graphs and a conjecture of Scott

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F16%3A10333128" target="_blank" >RIV/00216208:11320/16:10333128 - isvavai.cz</a>
Result on the web
<a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p30" target="_blank" >http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p30</a>
DOI - Digital Object Identifier
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Result language
angličtina
Original language name
Restricted frame graphs and a conjecture of Scott
Original language description
Scott proved in 1997 that for any tree T, every graph with bounded clique number which does not contain any subdivision of T as an induced subgraph has bounded chromatic number. Scott also conjectured that the same should hold if T is replaced by any graph H. Pawlik et al. recently constructed a family of triangle free intersection graphs of segments in the plane with unbounded chromatic number (thereby disproving an old conjecture of Erdos). This shows that Scott's conjecture is false whenever H is obtained from a non-planar graph by subdividing every edge at least once. It remains interesting to decide which graphs H satisfy Scott's conjecture and which do not. In this paper, we study the construction of Pawlik et al. in more details to extract more counterexamples to Scott's conjecture. For example, we show that Scott's conjecture is false for any graph obtained from K-4 by subdividing every edge at least once. We also prove that if G is a 2-connected multigraph with no vertex contained in every cycle of G, then any graph obtained from G by subdividing every edge at least twice is a counterexample to Scott's conjecture.
Czech name
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Czech description
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Type
J<sub>x</sub> - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
IN - Informatics
OECD FORD branch
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Project
<a href="/en/project/LL1201" target="_blank" >LL1201: Complex Structures: Regularities in Combinatorics and Discrete Mathematics</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year
2016
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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