Improvements on Hippchen's conjecture - Cho, Eun Kyung; Choi, Ilkyoo; Park, Boram (Discrete Mathematics, 2022)
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.
Keywords: Hippchen's conjecture; k-connected graph; Longest path