Let G be a k-connected graph on n vertices. Hippchen's Conjecture (2008) states that two longest paths in G share at least k vertices. Gutiérrez (2020) recently proved the conjecture when k≤4 or [Formula presented]. We improve upon both results; namely, we show that two longest paths in G share at least k vertices when k=5 or [Formula presented]. This completely resolves two conjectures by Gutiérrez in the affirmative.
Eun-Kyung Cho was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education ( NRF-2020R1I1A1A0105858711 ). Ilkyoo Choi was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education ( NRF-2018R1D1A1B07043049 ), and also by the Hankuk University of Foreign Studies Research Fund . Boram Park was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning ( NRF-2018R1C1B6003577 ).
