Abstract
As mentioned in the introduction, the aim of this chapter is to develop bijections which would enable a systematic characterization of the discrete spaces obtained by gluing bubbles according of their mean curvature. In particular, as detailed in Sect. 2.5, we are interested in identifying and counting the spaces which maximize the number of \((D-2)\)-cells at fixed number of D-cells. This is summarized in the guideline in Sect. 2.5.3. Our criteria for the bijection is therefore that it should keep track of the number of \((D-2)\)-cells, and map configurations to recognizable combinatorial families. We come to the general bijection step-by-step, by first looking at a simple case, then applying Tutte’s bijection, and then generalizing to any kind of colored graph.
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Notes
- 1.
As Eric Fusy suggested, it would actually be more natural to apply the version of Tutte’s bijection between bicolored maps and bipartite maps, instead of taking the dual first and applying the version of the bijection between bipartite maps and bipartite maps.
- 2.
In the usual graph theoretical sense.
- 3.
The permutation of color i is made explicit in \(B\) by keeping only edges of color i.
- 4.
i.e. as a formal power series in the coupling constant \(\lambda \).
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Lionni, L. (2018). Bijective Methods. In: Colored Discrete Spaces. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-96023-4_3
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