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Not All Saturated 3-Forests Are Tight
A basic statement in graph theory is that every inclusion-maximal forest is
connected, i.e. a tree. Using a definiton for higher dimensional forests by
Graham and Lovasz and the connectivity-related notion of tightness for
hypergraphs introduced by Arocha, Bracho and Neumann-Lara in, we provide an
example of a saturated, i.e. inclusion-maximal 3-forest that is not tight. This
resolves an open problem posed by Strausz
The -rainbow index of random graphs
A tree in an edge colored graph is said to be a rainbow tree if no two edges
on the tree share the same color. Given two positive integers , with
, the \emph{-rainbow index} of is the
minimum number of colors needed in an edge-coloring of such that for any
set of vertices of , there exist internally disjoint rainbow
trees connecting . This concept was introduced by Chartrand et. al., and
there have been very few related results about it. In this paper, We establish
a sharp threshold function for and
respectively, where and are
the usually defined random graphs.Comment: 7 pages. arXiv admin note: substantial text overlap with
arXiv:1212.6845, arXiv:1310.278
Graphs with 3-rainbow index and
Let be a nontrivial connected graph with an edge-coloring
, where adjacent edges
may be colored the same. A tree in is a if no two edges
of receive the same color. For a vertex set , the tree
connecting in is called an -tree. The minimum number of colors that
are needed in an edge-coloring of such that there is a rainbow -tree for
each -set of is called the -rainbow index of , denoted by
. In \cite{Zhang}, they got that the -rainbow index of a tree is
and the -rainbow index of a unicyclic graph is or . So
there is an intriguing problem: Characterize graphs with the -rainbow index
and . In this paper, we focus on , and characterize the graphs
whose 3-rainbow index is and , respectively.Comment: 14 page
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