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    Bi-traceable graphs, the intersection of three longest paths and Hippchen's conjecture
    (Cornell University, 2021-01-19)
    Let $P,Q$ be longest paths in a simple graph. We analyze the possible connections between the components of $P\cup Q\setminus (V(P)\cap V(Q))$ and introduce the notion of a bi-traceable graph. We use the results for all the possible configurations of the intersection points when $\#V(P)\cap V(Q)\le 5$ in order to prove that if the intersection of three longest paths $P,Q,R$ is empty, then $\#(V(P)\cap V(Q))\ge 6$. We also prove Hippchen's conjecture for $k\le 6$: If a graph $G$ is $k$-connected for $k\le 6$, and $P$ and $Q$ are longest paths in $G$, then $\#(V(P)\cap V(Q))\ge 6$.
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    On some indices of foliations and applications
    (Springer Science and Business Media Deutschland GmbH, 2026-03-01)
    In this paper, we establish a relationship between the Milnor number, the χ-number and the Tjurina number of a foliation with respect to an effective balanced divisor of separatrices. Moreover, using the Gómez-Mont–Seade–Verjovsky index, we prove that the difference between the multiplicity and the Tjurina number of a foliation with respect to a reduced curve is independent of the foliation. We also derive a local formula for the Tjurina number of a foliation with respect to a reduced curve. From a global point of view, these results lead to the following consequences: We provide a new proof of a global result regarding the multiplicity of a foliation due to Cerveau–Lins Neto and a new proof of a Soares’s inequality for the sum of the Milnor number of an invariant curve of a foliation. Additionally, we obtain bounds for the global Tjurina number of a foliation on the complex projective plane. Finally, we provide an answer to the conjecture posed by Alcántara and Mozo-Fernández about foliations on the complex projective plane having a unique singularity.
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    Number of homogeneous components of counterexamples to the Dixmier conjecture
    (Taylor and Francis Ltd., 2025)
    Assume that P and Q are elements of A1 satisfying [𝑃,𝑄]=1. The Dixmier Conjecture for A1 says that they always generate A1. We show that if P is a sum of not more than 4 homogeneous elements of A1 then P and Q generate A1, which generalizes the main result in [Citation10].
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