5,499 research outputs found
Higher melonic theories
We classify a large set of melonic theories with arbitrary -fold
interactions, demonstrating that the interaction vertices exhibit a range of
symmetries, always of the form for some , which may be .
The number of different theories proliferates quickly as increases above
and is related to the problem of counting one-factorizations of complete
graphs. The symmetries of the interaction vertex lead to an effective
interaction strength that enters into the Schwinger-Dyson equation for the
two-point function as well as the kernel used for constructing higher-point
functions.Comment: 43 pages, 12 figure
The Hamilton-Waterloo Problem with even cycle lengths
The Hamilton-Waterloo Problem HWP asks for a
2-factorization of the complete graph or , the complete graph with
the edges of a 1-factor removed, into -factors and
-factors, where . In the case that and are both
even, the problem has been solved except possibly when
or when and are both odd, in which case necessarily . In this paper, we develop a new construction that creates
factorizations with larger cycles from existing factorizations under certain
conditions. This construction enables us to show that there is a solution to
HWP for odd and whenever the obvious
necessary conditions hold, except possibly if ; and
; ; or . This result almost completely
settles the existence problem for even cycles, other than the possible
exceptions noted above
Tree-like properties of cycle factorizations
We provide a bijection between the set of factorizations, that is, ordered
(n-1)-tuples of transpositions in whose product is (12...n),
and labelled trees on vertices. We prove a refinement of a theorem of
D\'{e}nes that establishes new tree-like properties of factorizations. In
particular, we show that a certain class of transpositions of a factorization
correspond naturally under our bijection to leaf edges of a tree. Moreover, we
give a generalization of this fact.Comment: 10 pages, 3 figure
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