290 research outputs found
Ehrhart clutters: Regularity and Max-Flow Min-Cut
If C is a clutter with n vertices and q edges whose clutter matrix has column
vectors V={v1,...,vq}, we call C an Ehrhart clutter if {(v1,1),...,(vq,1)} is a
Hilbert basis. Letting A(P) be the Ehrhart ring of P=conv(V), we are able to
show that if A is the clutter matrix of a uniform, unmixed MFMC clutter C, then
C is an Ehrhart clutter and in this case we provide sharp bounds on the
Castelnuovo-Mumford regularity of A(P). Motivated by the Conforti-Cornuejols
conjecture on packing problems, we conjecture that if C is both ideal and the
clique clutter of a perfect graph, then C has the MFMC property. We prove this
conjecture for Meyniel graphs, by showing that the clique clutters of Meyniel
graphs are Ehrhart clutters. In much the same spirit, we provide a simple proof
of our conjecture when C is a uniform clique clutter of a perfect graph. We
close with a generalization of Ehrhart clutters as it relates to total dual
integrality.Comment: Electronic Journal of Combinatorics, to appea
Meyniel's conjecture holds for random graphs
In the game of cops and robber, the cops try to capture a robber moving on
the vertices of the graph. The minimum number of cops required to win on a
given graph is called the cop number of . The biggest open conjecture in
this area is the one of Meyniel, which asserts that for some absolute constant
, the cop number of every connected graph is at most .
In this paper, we show that Meyniel's conjecture holds asymptotically almost
surely for the binomial random graph. We do this by first showing that the
conjecture holds for a general class of graphs with some specific
expansion-type properties. This will also be used in a separate paper on random
-regular graphs, where we show that the conjecture holds asymptotically
almost surely when .Comment: revised versio
Independent Sets in Graphs with an Excluded Clique Minor
Let be a graph with vertices, with independence number , and
with with no -minor for some . It is proved that
Coloring vertices of a graph or finding a Meyniel obstruction
A Meyniel obstruction is an odd cycle with at least five vertices and at most
one chord. A graph is Meyniel if and only if it has no Meyniel obstruction as
an induced subgraph. Here we give a O(n^2) algorithm that, for any graph, finds
either a clique and coloring of the same size or a Meyniel obstruction. We also
give a O(n^3) algorithm that, for any graph, finds either aneasily recognizable
strong stable set or a Meyniel obstruction
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