9,745 research outputs found
Vertex decomposable graphs, codismantlability, Cohen-Macaulayness and Castelnuovo-Mumford regularity
We call a (simple) graph G codismantlable if either it has no edges or else
it has a codominated vertex x, meaning that the closed neighborhood of x
contains that of one of its neighbor, such that G-x codismantlable. We prove
that if G is well-covered and it lacks induced cycles of length four, five and
seven, than the vertex decomposability, codismantlability and
Cohen-Macaulayness for G are all equivalent. The rest deals with the
computation of Castelnuovo-Mumford regularity of codismantlable graphs. Note
that our approach complements and unifies many of the earlier results on
bipartite, chordal and very well-covered graphs
The fractional chromatic number of triangle-free subcubic graphs
Heckman and Thomas conjectured that the fractional chromatic number of any
triangle-free subcubic graph is at most 14/5. Improving on estimates of Hatami
and Zhu and of Lu and Peng, we prove that the fractional chromatic number of
any triangle-free subcubic graph is at most 32/11 (which is roughly 2.909)
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