1,165 research outputs found
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
Monochromatic cycle covers in random graphs
A classic result of Erd\H{o}s, Gy\'arf\'as and Pyber states that for every
coloring of the edges of with colors, there is a cover of its vertex
set by at most vertex-disjoint monochromatic cycles. In
particular, the minimum number of such covering cycles does not depend on the
size of but only on the number of colors. We initiate the study of this
phenomena in the case where is replaced by the random graph . Given a fixed integer and , we
show that with high probability the random graph has
the property that for every -coloring of the edges of , there is a
collection of monochromatic cycles covering all the
vertices of . Our bound on is close to optimal in the following sense:
if , then with high probability there are colorings of
such that the number of monochromatic cycles needed to
cover all vertices of grows with .Comment: 24 pages, 1 figure (minor changes, added figure
Ramified rectilinear polygons: coordinatization by dendrons
Simple rectilinear polygons (i.e. rectilinear polygons without holes or
cutpoints) can be regarded as finite rectangular cell complexes coordinatized
by two finite dendrons. The intrinsic -metric is thus inherited from the
product of the two finite dendrons via an isometric embedding. The rectangular
cell complexes that share this same embedding property are called ramified
rectilinear polygons. The links of vertices in these cell complexes may be
arbitrary bipartite graphs, in contrast to simple rectilinear polygons where
the links of points are either 4-cycles or paths of length at most 3. Ramified
rectilinear polygons are particular instances of rectangular complexes obtained
from cube-free median graphs, or equivalently simply connected rectangular
complexes with triangle-free links. The underlying graphs of finite ramified
rectilinear polygons can be recognized among graphs in linear time by a
Lexicographic Breadth-First-Search. Whereas the symmetry of a simple
rectilinear polygon is very restricted (with automorphism group being a
subgroup of the dihedral group ), ramified rectilinear polygons are
universal: every finite group is the automorphism group of some ramified
rectilinear polygon.Comment: 27 pages, 6 figure
-WORM colorings of graphs: Lower chromatic number and gaps in the chromatic spectrum
A -WORM coloring of a graph is an assignment of colors to the
vertices in such a way that the vertices of each -subgraph of get
precisely two colors. We study graphs which admit at least one such
coloring. We disprove a conjecture of Goddard et al. [Congr. Numer., 219 (2014)
161--173] who asked whether every such graph has a -WORM coloring with two
colors. In fact for every integer there exists a -WORM colorable
graph in which the minimum number of colors is exactly . There also exist
-WORM colorable graphs which have a -WORM coloring with two colors
and also with colors but no coloring with any of colors. We
also prove that it is NP-hard to determine the minimum number of colors and
NP-complete to decide -colorability for every (and remains
intractable even for graphs of maximum degree 9 if ). On the other hand,
we prove positive results for -degenerate graphs with small , also
including planar graphs. Moreover we point out a fundamental connection with
the theory of the colorings of mixed hypergraphs. We list many open problems at
the end.Comment: 18 page
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