5 research outputs found
Separating path systems
We study separating systems of the edges of a graph where each member of the
separating system is a path. We conjecture that every -vertex graph admits a
separating path system of size and prove this in certain interesting
special cases. In particular, we establish this conjecture for random graphs
and graphs with linear minimum degree. We also obtain tight bounds on the size
of a minimal separating path system in the case of trees.Comment: 21 pages, fixed misprints, Journal of Combinatoric
Sublinear expanders and their applications
In this survey we aim to give a comprehensive overview of results using
sublinear expanders. The term sublinear expanders refers to a variety of
definitions of expanders, which typically are defined to be graphs such
that every not-too-small and not-too-large set of vertices has
neighbourhood of size at least , where is a function of
and . This is in contrast with linear expanders, where is
typically a constant. :We will briefly describe proof ideas of some of the
results mentioned here, as well as related open problems.Comment: 39 pages, 15 figures. This survey will appear in `Surveys in
Combinatorics 2024' (the proceedings of the 30th British Combinatorial
Conference