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Large independent sets in triangle-free cubic graphs: beyond planarity

par Cames Van Batenburg, Wouter ;Goedgebeur, Jan;Joret, Gwenaël
Référence Advances in combinatorics, P7
Publication Publié, 2020-12-01

Article révisé par les pairs

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Résumé :

Every n-vertex planar triangle-free graph with maximum degree at most 3 has an independent set of size at least38 n. This was first conjectured by Albertson, Bollobás and Tucker, and was later proved by Heckman and Thomas. Fraughnaugh and Locke conjectured that the planarity requirement could be relaxed into just forbidding a few specific nonplanar subgraphs: They described a family F of six nonplanar graphs (each of order at most 22) and conjectured that every n-vertex triangle-free graph with maximum degree at most 3 having no subgraph isomorphic to a member of F has an independent set of size at least38 n. In this paper, we prove this conjecture. As a corollary, we obtain that every 2-connected n-vertex triangle-free graph with maximum degree at most 3 has an independent set of size at least38 n, with the exception of the six graphs in F. This confirms a conjecture made independently by Bajnok and Brinkmann, and by Fraughnaugh and Locke.