An asymptotic version of a conjecture by Enomoto and Ota

Abstract

AbstractIn 2000, Enomoto and Ota [J Graph Theory 34 (2000), 163–169] stated the following conjecture. Let G be a graph of order n, and let n1, n2, …, nk be positive integers with . If σ2(G)≥n+ k−1, then for any k distinct vertices x1, x2, …, xk in G, there exist vertex disjoint paths P1, P2, …, Pk such that |Pi|=ni and xi is an endpoint of Pi for every i, 1≤i≤k. We prove an asymptotic version of this conjecture in the following sense. For every k positive real numbers γ1, …, γk with , and for every ε>0, there exists n0 such that for every graph G of order n≥n0 with σ2(G)≥n+ k−1, and for every choice of k vertices x1, …, xk∈V(G), there exist vertex disjoint paths P1, …, Pk in G such that , the vertex xi is an endpoint of the path Pi, and (γi−ε)n<|Pi|<(γi + ε)n for every i, 1≤i≤k. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 37–51, 2010