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On the Average Genus of a Graph
Not all rational numbers are possibilities for the average genus of an individual graph. The smallest such numbers are determined, and varied examples are constructed to demonstrate that a single value of average genus can be shared by arbitrarily many different graphs. It is proved that the number one is a limit point of the set of possible values for average genus and that the complete graph K4 is the only 3-connected graph whose average genus is less than one. Several problems for future study are suggested
Random 2-cell embeddings of multistars
Random 2-cell embeddings of a given graph are obtained by choosing a
random local rotation around every vertex. We analyze the expected number of
faces, , of such an embedding which is equivalent to studying
its average genus. So far, tight results are known for two families called
monopoles and dipoles. We extend the dipole result to a more general family
called multistars, i.e., loopless multigraphs in which there is a vertex
incident with all the edges. In particular, we show that the expected number of
faces of every multistar with nonleaf edges lies in an interval of length
centered at the expected number of faces of an -edge dipole.
This allows us to derive bounds on for any given graph in
terms of vertex degrees. We conjecture that for any
simple -vertex graph .Comment: 15 pages, 2 figures. Accepted to European conference on
combinatorics, graph theory and applications (EUROCOMB) 202
Random Embeddings of Graphs: The Expected Number of Faces in Most Graphs is Logarithmic
A random 2-cell embedding of a connected graph in some orientable surface
is obtained by choosing a random local rotation around each vertex. Under this
setup, the number of faces or the genus of the corresponding 2-cell embedding
becomes a random variable. Random embeddings of two particular graph classes --
those of a bouquet of loops and those of parallel edges connecting two
vertices -- have been extensively studied and are well-understood. However,
little is known about more general graphs despite their important connections
with central problems in mainstream mathematics and in theoretical physics (see
[Lando & Zvonkin, Springer 2004]). There are also tight connections with
problems in computing (random generation, approximation algorithms). The
results of this paper, in particular, explain why Monte Carlo methods (see,
e.g., [Gross & Tucker, Ann. NY Acad. Sci 1979] and [Gross & Rieper, JGT 1991])
cannot work for approximating the minimum genus of graphs.
In his breakthrough work ([Stahl, JCTB 1991] and a series of other papers),
Stahl developed the foundation of "random topological graph theory". Most of
his results have been unsurpassed until today. In our work, we analyze the
expected number of faces of random embeddings (equivalently, the average genus)
of a graph . It was very recently shown [Campion Loth & Mohar, arXiv 2022]
that for any graph , the expected number of faces is at most linear. We show
that the actual expected number of faces is usually much smaller. In
particular, we prove the following results:
1) , for
sufficiently large. This greatly improves Stahl's upper bound for
this case.
2) For random models containing only graphs, whose maximum
degree is at most , we show that the expected number of faces is
.Comment: 44 pages, 6 figure