An Asymptotic Version of a Conjecture by Enomoto and OtaJournal of Graph Theory (2010)
In 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.
- Graph decomposition,
- Degree sum
Publication DateMay, 2010
Citation InformationColton Magnant and Daniel M. Martin. "An Asymptotic Version of a Conjecture by Enomoto and Ota" Journal of Graph Theory Vol. 64 Iss. 1 (2010) p. 37 - 51 ISSN: 1097-0118
Available at: http://works.bepress.com/colton_magnant/8/