Abstract
The CHY construction naturally associates a vector in ℝ(n−3)! to every 2- regular graph with n vertices. Partial amplitudes in the biadjoint scalar theory are given by the inner product of vectors associated with a pair of cycles. In this work we study the problem of extending the computation to pairs of arbitrary 2-regular graphs. This requires the construction of compatible cycles, i.e. cycles such that their union with a 2-regular graph admits a Hamiltonian decomposition. We prove that there are at least (n − 2)!/4 such cycles for any 2-regular graph. We also find a connection to breakpoint graphs when the initial 2-regular graph only has double edges. We end with a comparison of the lower bound on the number of randomly selected cycles needed to generate a basis of ℝ(n−3)!, using the super Catalan numbers and our lower bound for compatible cycles.
Highlights
In this work we study the problem of extending the computation to pairs of arbitrary 2-regular graphs. This requires the construction of compatible cycles, i.e. cycles such that their union with a 2-regular graph admits a Hamiltonian decomposition
We end with a comparison of the lower bound on the number of randomly selected cycles needed to generate a basis of R(n−3)!, using the super Catalan numbers and our lower bound for compatible cycles
Given a 2-regular graph G, a compatible cycle to G is a cycle C such that the 4-regular graph obtained by the edge-disjoint union G ∪ C admits a hamiltonian decomposition, i.e., G ∪ C = C1 ∪ C2 where C1 and C2 are both cycles on the same vertex set as G
Summary
We end the introduction with a short review of graph theory terminology. Readers are encourage to skip this in a first reading and only use it if needed. A graph is loopless if it has no edge with both ends at the same vertex. Given a 2k-regular graph G, a hamiltonian decomposition of G, when it exists, is a decomposition of the edges of G into k disjoint hamiltonian cycles: G = C1 ∪ C2 ∪ · · · ∪ Ck with each Cj a cycle on the same vertex set as G. We will use the notion of perfect matching on a vertex set (without the requirement of being a subgraph of some G), meaning a 1-regular graph on that vertex set. Given a perfect matching M in a graph G, and a vertex v of G, the M -neighbour of v is the vertex connected to v by an edge of M. For more graph theory background the reader is referred to [17] or [18]
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