9,064 research outputs found
Rainbow Matchings and Hamilton Cycles in Random Graphs
Let be drawn uniformly from all -uniform, -partite
hypergraphs where each part of the partition is a disjoint copy of . We
let HP^{(\k)}_{n,m,k} be an edge colored version, where we color each edge
randomly from one of \k colors. We show that if \k=n and where
is sufficiently large then w.h.p. there is a rainbow colored perfect
matching. I.e. a perfect matching in which every edge has a different color. We
also show that if is even and where is sufficiently large
then w.h.p. there is a rainbow colored Hamilton cycle in . Here
denotes a random edge coloring of with colors.
When is odd, our proof requires m=\om(n\log n) for there to be a rainbow
Hamilton cycle.Comment: We replaced graphs by k-uniform hypergraph
Partitioning Perfect Graphs into Stars
The partition of graphs into "nice" subgraphs is a central algorithmic
problem with strong ties to matching theory. We study the partitioning of
undirected graphs into same-size stars, a problem known to be NP-complete even
for the case of stars on three vertices. We perform a thorough computational
complexity study of the problem on subclasses of perfect graphs and identify
several polynomial-time solvable cases, for example, on interval graphs and
bipartite permutation graphs, and also NP-complete cases, for example, on grid
graphs and chordal graphs.Comment: Manuscript accepted to Journal of Graph Theor
Local Maximum Stable Sets Greedoids Stemmed from Very Well-Covered Graphs
A maximum stable set in a graph G is a stable set of maximum cardinality. S
is called a local maximum stable set of G if S is a maximum stable set of the
subgraph induced by the closed neighborhood of S. A greedoid (V,F) is called a
local maximum stable set greedoid if there exists a graph G=(V,E) such that its
family of local maximum stable sets coinsides with (V,F). It has been shown
that the family local maximum stable sets of a forest T forms a greedoid on its
vertex set. In this paper we demonstrate that if G is a very well-covered
graph, then its family of local maximum stable sets is a greedoid if and only
if G has a unique perfect matching.Comment: 12 pages, 12 figure
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