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Partial-dual Euler-genus distributions for bouquets with small Euler genus. (English) Zbl 1492.05035

Summary: For an arbitrary ribbon graph \(G\), the partial-dual Euler-genus polynomial of \(G\) is a generating function that enumerates partial duals of \(G\) by Euler genus. When \(G\) is an orientable ribbon graph, the partial-dual orientable genus polynomial of \(G\) is a generating function that enumerates partial duals of \(G\) by orientable genus. J. L. Gross et al. [Eur. J. Comb. 86, Article ID 103084, 20 p. (2020; Zbl 1437.05060)] inaugurated these partial-dual Euler-genus and orientable genus distribution problems in 2020. A bouquet is a one-vertex ribbon graph. Given a ribbon graph \(G\), its partial-dual Euler-genus polynomial is the same as that of some bouquet; this motivates our focus on bouquets. We obtain the partial-dual Euler-genus polynomials for all the bouquets with Euler genus at most two.

MSC:

05C10 Planar graphs; geometric and topological aspects of graph theory
05C30 Enumeration in graph theory
05C31 Graph polynomials
57M15 Relations of low-dimensional topology with graph theory
11B68 Bernoulli and Euler numbers and polynomials

Citations:

Zbl 1437.05060
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References:

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