Enomoto and Ota’s Conjecture Holds for Large Graphs

Abstract

In 2000, Enomoto and Ota conjectured that if a graph G satisfies $$\sigma _{2}(G) \ge |G| + k - 1$$ , then for any set of k vertices $$v_{1}, \ldots , v_{k}$$ and for any positive integers $$n_{1}, \ldots , n_{k}$$ with $$\sum n_{i} = |G|$$ , there exists a partition of V(G) into k paths $$P_{1}, \ldots , P_{k}$$ 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.