4,030 research outputs found
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
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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