424 research outputs found
Combinatorial symbolic powers
Symbolic powers are studied in the combinatorial context of monomial ideals.
When the ideals are generated by quadratic squarefree monomials, the generators
of the symbolic powers are obstructions to vertex covering in the associated
graph and its blowups. As a result, perfect graphs play an important role in
the theory, dual to the role played by perfect graphs in the theory of secants
of monomial ideals. We use Gr\"obner degenerations as a tool to reduce
questions about symbolic powers of arbitrary ideals to the monomial case. Among
the applications are a new, unified approach to the Gr\"obner bases of symbolic
powers of determinantal and Pfaffian ideals.Comment: 29 pages, 3 figures, Positive characteristic results incorporated
into main body of pape
Nordhaus-Gaddum for Treewidth
We prove that for every graph with vertices, the treewidth of
plus the treewidth of the complement of is at least . This bound is
tight
Minimal classes of graphs of unbounded clique-width defined by finitely many forbidden induced subgraphs
We discover new hereditary classes of graphs that are minimal (with respect
to set inclusion) of unbounded clique-width. The new examples include split
permutation graphs and bichain graphs. Each of these classes is characterised
by a finite list of minimal forbidden induced subgraphs. These, therefore,
disprove a conjecture due to Daligault, Rao and Thomasse from 2010 claiming
that all such minimal classes must be defined by infinitely many forbidden
induced subgraphs.
In the same paper, Daligault, Rao and Thomasse make another conjecture that
every hereditary class of unbounded clique-width must contain a labelled
infinite antichain. We show that the two example classes we consider here
satisfy this conjecture. Indeed, they each contain a canonical labelled
infinite antichain, which leads us to propose a stronger conjecture: that every
hereditary class of graphs that is minimal of unbounded clique-width contains a
canonical labelled infinite antichain.Comment: 17 pages, 7 figure
Research in structural graph theory
Issued as final reportNational Science Foundation (U.S.
Some results on triangle partitions
We show that there exist efficient algorithms for the triangle packing
problem in colored permutation graphs, complete multipartite graphs,
distance-hereditary graphs, k-modular permutation graphs and complements of
k-partite graphs (when k is fixed). We show that there is an efficient
algorithm for C_4-packing on bipartite permutation graphs and we show that
C_4-packing on bipartite graphs is NP-complete. We characterize the cobipartite
graphs that have a triangle partition
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