Restricted Frame Graphs and a Conjecture of Scott

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 trianglefree 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 K4 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. ∗Partially supported by ANR Project MACARON (anr-13-js02-0002). †Partially supported by ANR Project Heredia (anr-10-jcjc-0204-01), ANR Project Stint (anr-13bs02-0007), and LabEx PERSYVAL-Lab (anr-11-labx-0025-01). ‡Partially supported by grant ERCCZ LL-1201, by the European Associated Laboratory “Structures in Combinatorics” (LEA STRUCO), and by ANR Project Stint anr-13-bs02-0007. the electronic journal of combinatorics 23(1) (2016), #P1.30 1