1,315 research outputs found
Two remarks on retracts of graph products
AbstractLet H be a bipartie graph and let Gn be the Mycielski graph with χ(G) = n, n ⩾ 4. Then the chromatic number of the strong product of Gn by H is at most 2n − 2. We use this result to show that there exist strong products of graphs in which a projection of a retract onto a factor is not a retract of the factor. We also show that in the Cartesian product of graphs G and H, any tringles of G transfer in H, whenever G and H are connected and G is strongly-triangulated, weakly-triangulated or four-cycle free
On retracts, absolute retracts, and folds in cographs
Let G and H be two cographs. We show that the problem to determine whether H
is a retract of G is NP-complete. We show that this problem is fixed-parameter
tractable when parameterized by the size of H. When restricted to the class of
threshold graphs or to the class of trivially perfect graphs, the problem
becomes tractable in polynomial time. The problem is also soluble when one
cograph is given as an induced subgraph of the other. We characterize absolute
retracts of cographs.Comment: 15 page
Special Lagrangian torus fibrations of complete intersection Calabi-Yau manifolds: a geometric conjecture
For complete intersection Calabi-Yau manifolds in toric varieties, Gross and
Haase-Zharkov have given a conjectural combinatorial description of the special
Lagrangian torus fibrations whose existence was predicted by Strominger, Yau
and Zaslow. We present a geometric version of this construction, generalizing
an earlier conjecture of the first author.Comment: 23 pagers, 10 figure
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