54,051 research outputs found
Equitable partition of graphs into induced forests
An equitable partition of a graph is a partition of the vertex-set of
such that the sizes of any two parts differ by at most one. We show that every
graph with an acyclic coloring with at most colors can be equitably
partitioned into induced forests. We also prove that for any integers
and , any -degenerate graph can be equitably
partitioned into induced forests.
Each of these results implies the existence of a constant such that for
any , any planar graph has an equitable partition into induced
forests. This was conjectured by Wu, Zhang, and Li in 2013.Comment: 4 pages, final versio
Isolating highly connected induced subgraphs
We prove that any graph of minimum degree greater than has a
-connected induced subgraph such that the number of vertices of
that have neighbors outside of is at most . This generalizes a
classical result of Mader, which states that a high minimum degree implies the
existence of a highly connected subgraph. We give several variants of our
result, and for each of these variants, we give asymptotics for the bounds. We
also we compute optimal values for the case when . Alon, Kleitman, Saks,
Seymour, and Thomassen proved that in a graph of high chromatic number, there
exists an induced subgraph of high connectivity and high chromatic number. We
give a new proof of this theorem with a better bound
Topological and geometrical restrictions, free-boundary problems and self-gravitating fluids
Let (P1) be certain elliptic free-boundary problem on a Riemannian manifold
(M,g). In this paper we study the restrictions on the topology and geometry of
the fibres (the level sets) of the solutions f to (P1). We give a technique
based on certain remarkable property of the fibres (the analytic representation
property) for going from the initial PDE to a global analytical
characterization of the fibres (the equilibrium partition condition). We study
this analytical characterization and obtain several topological and geometrical
properties that the fibres of the solutions must possess, depending on the
topology of M and the metric tensor g. We apply these results to the classical
problem in physics of classifying the equilibrium shapes of both Newtonian and
relativistic static self-gravitating fluids. We also suggest a relationship
with the isometries of a Riemannian manifold.Comment: 36 pages. In this new version the analytic representation hypothesis
is proved. Please address all correspondence to D. Peralta-Sala
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