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Constructing Trees in Graphs whose Complement has no K2,s

Published online by Cambridge University Press:  06 September 2002

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

We show that, if G is a graph of order n with maximal degree Δ(G) and minimal degree δ(G) whose complement contains no K2,s, s [ges ] 2, then G contains every tree T of order n−s+1 whose maximal degree is at most Δ(G) and whose vertex of second-largest degree is at most δ(G). We then show that this result implies that special cases of two conjectures are true. We verify that the Erdös–Sós conjecture, which states that a graph whose average degree is larger than k−1 contains every tree of order k+1, is true for graphs whose complement does not contain a K2,4, and the Komlós–Sós conjecture, which states that every graph of median degree at least k contains every tree of order k+1, is true for graphs whose complement does not contain a K2,3.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 11 , Issue 4 , July 2002 , pp. 343 - 347
Copyright
2002 Cambridge University Press

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