1 research outputs found
On the balanced decomposition number
A {\em balanced coloring} of a graph means a triple of
mutually disjoint subsets of the vertex-set such that and . A {\em balanced decomposition} associated with
the balanced coloring of is defined as a
partition of (for some ) such that, for
every , the subgraph of is connected and
. Then the {\em balanced decomposition number}
of a graph is defined as the minimum integer such that, for every
balanced coloring of , there exists a
balanced decomposition whose every element
has at most vertices. S. Fujita and H. Liu [\/SIAM
J. Discrete Math. 24, (2010), pp. 1597--1616\/] proved a nice theorem which
states that the balanced decomposition number of a graph is at most if
and only if is -connected. Unfortunately,
their proof is lengthy (about 10 pages) and complicated. Here we give an
immediate proof of the theorem. This proof makes clear a relationship between
balanced decomposition number and graph matching