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Ramsey-nice families of graphs
For a finite family of fixed graphs let be
the smallest integer for which every -coloring of the edges of the
complete graph yields a monochromatic copy of some . We
say that is -nice if for every graph with
and for every -coloring of there exists a
monochromatic copy of some . It is easy to see that if
contains no forest, then it is not -nice for any . It seems
plausible to conjecture that a (weak) converse holds, namely, for any finite
family of graphs that contains at least one forest, and for all
(or at least for infinitely many values of ),
is -nice. We prove several (modest) results in support of this
conjecture, showing, in particular, that it holds for each of the three
families consisting of two connected graphs with 3 edges each and observing
that it holds for any family containing a forest with at most 2
edges. We also study some related problems and disprove a conjecture by
Aharoni, Charbit and Howard regarding the size of matchings in regular
3-partite 3-uniform hypergraphs.Comment: 20 pages, 2 figure
Few Long Lists for Edge Choosability of Planar Cubic Graphs
It is known that every loopless cubic graph is 4-edge choosable. We prove the
following strengthened result.
Let G be a planar cubic graph having b cut-edges. There exists a set F of at
most 5b/2 edges of G with the following property. For any function L which
assigns to each edge of F a set of 4 colours and which assigns to each edge in
E(G)-F a set of 3 colours, the graph G has a proper edge colouring where the
colour of each edge e belongs to L(e).Comment: 14 pages, 1 figur
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