Enomoto and Ota's conjecture holds for large graphs

Abstract

In 2000, Enomoto and Ota conjectured that if a graph $G$ satisfies $\sigma_{2}(G) \geq n + k - 1$, then for any set of $k$ vertices $v_{1}, \dots, v_{k}$ and for any positive integers $n_{1}, \dots, n_{k}$ with $\sum n_{i} = |G|$, there exists a partition of $V(G)$ into $k$ paths $P_{1}, \dots, P_{k}$ such that $v_{i}$ is an end of $P_{i}$ and $|P_{i}| = n_{i}$ for all $i$. We prove this conjecture when $|G|$ is large. Our proof uses the Regularity Lemma along with several extremal lemmas, concluding with an absorbing argument to retrieve misbehaving vertices.