Geodesic
Restricted frame graphs and a conjecture of Scott
Jérémie Chalopin1 ; Louis Esperet2 ; Zhentao Li3 ; Patrice Ossona de Mendez4
1CNRS & Aix-Marseille Université
2CNRS & Univ. Grenoble Alpes
3École Normale Supérieure
4EHESS and Charles University
The electronic journal of combinatorics, Tome 23 (2016) no. 1
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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 Erdős). 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.
DOI : 10.37236/4424
Classification : 05C15, 05C75, 05C62
Mots-clés : chromatic number, subdivisions, chi-boundedness, Scott's conjecture, intersection graphs

Jérémie Chalopin  1   ; Louis Esperet  2   ; Zhentao Li  3   ; Patrice Ossona de Mendez  4

1 CNRS & Aix-Marseille Université
2 CNRS & Univ. Grenoble Alpes
3 École Normale Supérieure
4 EHESS and Charles University 
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     title = {Restricted frame graphs and a conjecture of {Scott}},
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     year = {2016},
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Jérémie Chalopin; Louis Esperet; Zhentao Li; Patrice Ossona de Mendez. Restricted frame graphs and a conjecture of Scott. The electronic journal of combinatorics, Tome 23 (2016) no. 1. doi: 10.37236/4424

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