Abstract
A long-standing conjecture asserts that there is a positive constant c such that every n-vertex graph without isolated vertices contains an induced subgraph with all degrees odd on at least cn vertices. Recently, Ferber and Krivelevich confirmed the conjecture with \(c\ge 10^{-4}\). However, this is far from optimal for special family of graphs. Scott proved that \(c\ge (2\chi )^{-1}\) for graphs with chromatic number \(\chi \ge 2\) and conjectured that \(c\ge \chi ^{-1}\). Partial tight bounds of c are also established by various authors for graphs such as trees, graphs with maximum degree 3 or \(K_4\)-minor-free graphs. In this paper, we further prove that \(c\ge 2/5\) for planar graphs with girth at least 7, and the bound is tight. We also show that \(c\le 1/3\) for general planar graphs and \(c\ge 1/3\) for planar graphs with girth at least 6.
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Acknowledgements
The authors would like to thank Prof. Jie Ma for bringing this topic to their attention. The authors are also grateful to Dr. Heng Li for helpful discussion on an earlier version of this manuscript.
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Research partially supported by NSFC (Grant No. 12001106), National Natural Science Foundation of Fujian Province (Grant No. 2021J05128) and Foundation of Fuzhou University (Grant No. GXRC-20059).
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Rao, M., Hou, J. & Zeng, Q. Odd Induced Subgraphs in Planar Graphs with Large Girth. Graphs and Combinatorics 38, 105 (2022). https://doi.org/10.1007/s00373-022-02499-7
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DOI: https://doi.org/10.1007/s00373-022-02499-7