Packing Trees into the Complete Graph

  • Authors:

  • Edward Dobson

  • Affiliations:

  • Department of Mathematics and Statistics, PO Drawer MA, Mississippi State University, Mississippi State, MS 39762, USA (e-mail: dobson@math.msstate.edu)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2002

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Abstract

Let c ⩽ 0.076122 and T1, T2,…, Tn be a sequence of trees such that ∣V(Ti)∣ ⩽ i−c(i−1). We prove that, if for each 1 ⩽ i ⩽ n there exists a vertex xi ∈ V(Ti) such that Ti−xi has at least (1−2c)(i−1) isolated vertices, then T1,…, Tn can be packed into Kn. We also prove that if T is a tree of order n+1−c′n, c′ ⩽ 1/25 (37−8 √21 ) ≈ 0.0135748, such that there exists a vertex x ∈ V(T) and T−x has at least n(1−2c′) isolated vertices, then 2n+1 copies of T may be packed into K2n+1.