155,216 research outputs found
On the Number of 1-Factors of Locally Finite Graphs
AbstractEvery infinite locally finite graph with exactly one 1-factor is at most 2-connected is shown. More generally a lower bound for the number of 1-factors in locally finite n-connected graphs is given
On locally rainbow colourings
Given a graph , let denote the smallest for which the
following holds. We can assign a -colouring of the edge set of
to each vertex in with the property that for any copy of in
, there is some such that every edge in has a different
colour in .
The study of this function was initiated by Alon and Ben-Eliezer. They
characterized the family of graphs for which is bounded and asked
whether it is true that for every other graph is polynomial. We show
that this is not the case and characterize the family of connected graphs
for which grows polynomially. Answering another question of theirs, we
also prove that for every , there is some
such that for all sufficiently large .
Finally, we show that the above problem is connected to the
Erd\H{o}s-Gy\'arf\'as function in Ramsey Theory, and prove a family of special
cases of a conjecture of Conlon, Fox, Lee and Sudakov by showing that for each
fixed the complete -uniform hypergraph can be edge-coloured
using a subpolynomial number of colours in such a way that at least colours
appear among any vertices.Comment: 12 page
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