Enomoto and Ota's conjecture holds for large graphs

Abstract

In 2000, Enomoto and Ota conjectured that if a graph $G$ satisfies ${\sigma }_2(G)\ge n+k-1$, 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.