240 research outputs found
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
Color the cycles
The cycles of length k in a complete graph on n vertices are colored in such a way that edge-disjoint cycles get distinct colors. The minimum number of colors is asymptotically determined. © 2013
A special case of Vu's conjecture: Coloring nearly disjoint graphs of bounded maximum degree
A collection of graphs is \textit{nearly disjoint} if every pair of them
intersects in at most one vertex. We prove that if are nearly
disjoint graphs of maximum degree at most , then the following holds. For
every fixed , if each vertex is contained in
at most of the graphs , then the (list) chromatic number
of is at most . This result confirms a special
case of a conjecture of Vu and generalizes Kahn's bound on the list chromatic
index of linear uniform hypergraphs of bounded maximum degree. In fact, this
result holds for the correspondence (or DP) chromatic number and thus implies a
recent result of Molloy, and we derive this result from a more general list
coloring result in the setting of `color degrees' that also implies a result of
Reed and Sudakov.Comment: 14 pages with one-page appendix; this version adds Theorem 1.5 due to
L. Postl
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