2,349 research outputs found
Counting dimer coverings on self-similar Schreier graphs
We study partition functions for the dimer model on families of finite graphs
converging to infinite self-similar graphs and forming approximation sequences
to certain well-known fractals. The graphs that we consider are provided by
actions of finitely generated groups by automorphisms on rooted trees, and thus
their edges are naturally labeled by the generators of the group. It is thus
natural to consider weight functions on these graphs taking different values
according to the labeling. We study in detail the well-known example of the
Hanoi Towers group , closely related to the Sierpi\'nski gasket.Comment: 29 pages. Final version, to appear in European Journal of
Combinatoric
Vertex covers by monochromatic pieces - A survey of results and problems
This survey is devoted to problems and results concerning covering the
vertices of edge colored graphs or hypergraphs with monochromatic paths, cycles
and other objects. It is an expanded version of the talk with the same title at
the Seventh Cracow Conference on Graph Theory, held in Rytro in September
14-19, 2014.Comment: Discrete Mathematics, 201
Vertex covering with monochromatic pieces of few colours
In 1995, Erd\H{o}s and Gy\'arf\'as proved that in every -colouring of the
edges of , there is a vertex cover by monochromatic paths of
the same colour, which is optimal up to a constant factor. The main goal of
this paper is to study the natural multi-colour generalization of this problem:
given two positive integers , what is the smallest number
such that in every colouring of the edges of with
colours, there exists a vertex cover of by
monochromatic paths using altogether at most different colours? For fixed
integers and as , we prove that , where is the chromatic number of
the Kneser gr aph . More generally, if one replaces by
an arbitrary -vertex graph with fixed independence number , then we
have , where this time around is the
chromatic number of the Kneser hypergraph . This
result is tight in the sense that there exist graphs with independence number
for which . This is in sharp
contrast to the case , where it follows from a result of S\'ark\"ozy
(2012) that depends only on and , but not on
the number of vertices. We obtain similar results for the situation where
instead of using paths, one wants to cover a graph with bounded independence
number by monochromatic cycles, or a complete graph by monochromatic
-regular graphs
Partitioning 3-colored complete graphs into three monochromatic cycles
We show in this paper that in every 3-coloring of the edges of Kn all but o(n)
of its vertices can be partitioned into three monochromatic cycles. From this, using
our earlier results, actually it follows that we can partition all the vertices into at
most 17 monochromatic cycles, improving the best known bounds. If the colors of
the three monochromatic cycles must be different then one can cover ( 3
4 â o(1))n
vertices and this is close to best possible
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