18,095 research outputs found
Maximising -Colourings of Graphs
For graphs and , an -colouring of is a map
such that . The number of -colourings of is denoted by .
We prove the following: for all graphs and , there is a
constant such that, if , the graph
maximises the number of -colourings among all
connected graphs with vertices and minimum degree . This answers a
question of Engbers.
We also disprove a conjecture of Engbers on the graph that maximises the
number of -colourings when the assumption of the connectivity of is
dropped.
Finally, let be a graph with maximum degree . We show that, if
does not contain the complete looped graph on vertices or as a
component and , then the following holds: for
sufficiently large, the graph maximises the number of
-colourings among all graphs on vertices with minimum degree .
This partially answers another question of Engbers
Critical Hardy--Sobolev Inequalities
We consider Hardy inequalities in , , with best constant
that involve either distance to the boundary or distance to a surface of
co-dimension , and we show that they can still be improved by adding a
multiple of a whole range of critical norms that at the extreme case become
precisely the critical Sobolev norm.Comment: 22 page
On the lattice of cotorsion theories
We discuss the lattice of cotorsion theories for abelian groups. First we
show that the sublattice of the well-studied rational cotorsion theories can be
identified with the well-known lattice of types. Using a recently developed
method for making Ext vanish we also prove that any power set together with the
ordinary set inclusion (and thus any poset) can be embedded into the lattice of
all cotorsion theories
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