40 research outputs found
On the Connectivity of Token Graphs of Trees
Let and be integers such that , and let be a
simple graph of order . The -token graph of is the graph
whose vertices are the -subsets of , where two vertices are adjacent
in whenever their symmetric difference is an edge of . In this
paper we show that if is a tree, then the connectivity of is equal
to the minimum degree of
Independence and matching numbers of some token graphs
Let be a graph of order and let . The -token
graph of , is the graph whose vertices are the -subsets of
, where two vertices are adjacent in whenever their symmetric
difference is an edge of . We study the independence and matching numbers of
. We present a tight lower bound for the matching number of
for the case in which has either a perfect matching or an almost perfect
matching. Also, we estimate the independence number for bipartite -token
graphs, and determine the exact value for some graphs.Comment: 16 pages, 4 figures. Third version is a major revision. Some proofs
were corrected or simplified. New references adde
The packing number of the double vertex graph of the path graph
Neil Sloane showed that the problem of determine the maximum size of a binary
code of constant weight 2 that can correct a single adjacent transposition is
equivalent to finding the packing number of a certain graph. In this paper we
solve this open problem by finding the packing number of the double vertex
graph (2-token graph) of a path graph. This double vertex graph is isomorphic
to the Sloane's graph. Our solution implies a conjecture of Rob Pratt about the
ordinary generating function of sequence A085680.Comment: 21 pages, 7 figures. V2: 22 pages, more figures added. V3. minor
corrections based on referee's comments. One figure corrected. The title "On
an error correcting code problem" has been change
The Maximum Chromatic Number of the Disjointness Graph of Segments on -point Sets in the Plane with
Let be a finite set of points in general position in the plane. The
disjointness graph of segments of is the graph whose vertices are
all the closed straight line segments with endpoints in , two of which are
adjacent in if and only if they are disjoint. As usual, we use
to denote the chromatic number of , and use to denote
the maximum taken over all sets of points in general
position in the plane. In this paper we show that if and only if
.Comment: 25 pages, 3 figure