40,333 research outputs found
On the choosability of claw-free perfect graphs
It has been conjectured that for every claw-free graph the choice number
of is equal to its chromatic number. We focus on the special case of this
conjecture where is perfect. Claw-free perfect graphs can be decomposed via
clique-cutset into two special classes called elementary graphs and peculiar
graphs. Based on this decomposition we prove that the conjecture holds true for
every claw-free perfect graph with maximum clique size at most
A Multipartite Hajnal-Szemer\'edi Theorem
The celebrated Hajnal-Szemer\'edi theorem gives the precise minimum degree
threshold that forces a graph to contain a perfect K_k-packing. Fischer's
conjecture states that the analogous result holds for all multipartite graphs
except for those formed by a single construction. Recently, we deduced an
approximate version of this conjecture from new results on perfect matchings in
hypergraphs. In this paper, we apply a stability analysis to the extremal cases
of this argument, thus showing that the exact conjecture holds for any
sufficiently large graph.Comment: Final version, accepted to appear in JCTB. 43 pages, 2 figure
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
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