4,148 research outputs found
Tropical covers of curves and their moduli spaces
We define the tropical moduli space of covers of a tropical line in the plane
as weighted abstract polyhedral complex, and the tropical branch map recording
the images of the simple ramifications. Our main result is the invariance of
the degree of the branch map, which enables us to give a tropical
intersection-theoretic definition of tropical triple Hurwitz numbers. We show
that our intersection-theoretic definition coincides with the one given by
Bertrand, Brugall\'e and Mikhalkin in the article "Tropical Open Hurwitz
numbers" where a Correspondence Theorem for Hurwitz numbers is proved. Thus we
provide a tropical intersection-theoretic justification for the multiplicities
with which a tropical cover has to be counted. Our method of proof is to
establish a local duality between our tropical moduli spaces and certain moduli
spaces of relative stable maps to the projective line.Comment: 24 pages, 10 figure
Polynomiality, Wall Crossings and Tropical Geometry of Rational Double Hurwitz Cycles
We study rational double Hurwitz cycles, i.e. loci of marked rational stable
curves admitting a map to the projective line with assigned ramification
profiles over two fixed branch points. Generalizing the phenomenon observed for
double Hurwitz numbers, such cycles are piecewise polynomial in the entries of
the special ramification; the chambers of polynomiality and wall crossings have
an explicit and "modular" description. A main goal of this paper is to
simultaneously carry out this investigation for the corresponding objects in
tropical geometry, underlining a precise combinatorial duality between
classical and tropical Hurwitz theory
Irreducible triangulations of surfaces with boundary
A triangulation of a surface is irreducible if no edge can be contracted to
produce a triangulation of the same surface. In this paper, we investigate
irreducible triangulations of surfaces with boundary. We prove that the number
of vertices of an irreducible triangulation of a (possibly non-orientable)
surface of genus g>=0 with b>=0 boundaries is O(g+b). So far, the result was
known only for surfaces without boundary (b=0). While our technique yields a
worse constant in the O(.) notation, the present proof is elementary, and
simpler than the previous ones in the case of surfaces without boundary
Cycle Double Covers and Semi-Kotzig Frame
Let be a cubic graph admitting a 3-edge-coloring
such that the edges colored by 0 and induce a Hamilton circuit
of and the edges colored by 1 and 2 induce a 2-factor . The graph is
semi-Kotzig if switching colors of edges in any even subgraph of yields a
new 3-edge-coloring of having the same property as . A spanning subgraph
of a cubic graph is called a {\em semi-Kotzig frame} if the contracted
graph is even and every non-circuit component of is a subdivision of
a semi-Kotzig graph.
In this paper, we show that a cubic graph has a circuit double cover if
it has a semi-Kotzig frame with at most one non-circuit component. Our result
generalizes some results of Goddyn (1988), and H\"{a}ggkvist and Markstr\"{o}m
[J. Combin. Theory Ser. B (2006)]
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