16 research outputs found
Subgraph statistics in subcritical graph classes
Let H be a fixed graph and math formula a subcritical graph class. In this paper we show that the number of occurrences of H (as a subgraph) in a graph in math formula of order n, chosen uniformly at random, follows a normal limiting distribution with linear expectation and variance. The main ingredient in our proof is the analytic framework developed by Drmota, Gittenberger and Morgenbesser to deal with infinite systems of functional equations [Drmota, Gittenberger, and Morgenbesser, Submitted]. As a case study, we obtain explicit expressions for the number of triangles and cycles of length 4 in the family of series-parallel graphs.Postprint (author's final draft
Subcritical graph classes containing all planar graphs
We construct minor-closed addable families of graphs that are subcritical and
contain all planar graphs. This contradicts (one direction of) a well-known
conjecture of Noy
Enumeration of chordal planar graphs and maps
We determine the number of labelled chordal planar graphs with n vertices, which is asymptotically for a constant and . We also determine the number of rooted simple chordal planar maps with n edges, which is asymptotically , where , , and s is an algebraic number of degree 12. The proofs are based on combinatorial decompositions and singularity analysis. Chordal planar graphs (or maps) are a natural example of a subcritical class of graphs in which the class of 3-connected graphs is relatively rich. The 3-connected members are precisely chordal triangulations, those obtained starting from by repeatedly adding vertices adjacent to an existing triangular face.We gratefully acknowledge earlier discussions on this project with Erkan Narmanli. M.N. was supported by grants MTM2017-82166-P and PID2020-113082GB-I00, the Severo Ochoa and MarĂa de Maeztu Program for Centers and Units of Excellence in R&D (CEX2020-001084-M). C.R. was supported by the grant Beatriu de PinĂłs BP2019, funded by the H2020 COFUND project No 801370 and AGAUR (the Catalan agency for management of university and research grants), and the grant PID2020-113082GB-I00 of the Spanish Ministry of Science and Innovation.Peer ReviewedPostprint (author's final draft
Enumeration of rooted 3-connected bipartite planar maps
We provide the first solution to the problem of counting rooted 3-connected
bipartite planar maps. Our starting point is the enumeration of bicoloured
planar maps according to the number of edges and monochromatic edges, following
Bernardi and Bousquet-M\'elou [J. Comb. Theory Ser. B, 101 (2011), 315-377].
The decomposition of a map into 2- and 3-connected components allows us to
obtain the generating functions of 2-and 3-connected bicoloured maps. Setting
to zero the variable marking monochromatic edges we obtain the generating
function of 3-connected bipartite maps, which is algebraic of degree 26. We
deduce from it an asymptotic estimate for the number of 3-connected bipartite
planar maps of the form , where and is an algebraic number of degree
10.Comment: 15 pages, 5 figure
Encoding and avoiding 2-connected patterns in polygon dissections and outerplanar graphs
Let Âż={d1,d2,...,dm} be a finite set of 2-connected patterns, i.e. graphs up to vertex relabelling. We study the generating function DÂż(z,u1,u2,...,um), which counts polygon dissections and marks subgraph copies of di with the variable ui. We prove that this is always algebraic, through an explicit combinatorial decomposition depending on Âż. The decomposition also gives a defining system for DÂż(z,0), which encodes polygon dissections that restrict these patterns as subgraphs. In this way, we are able to extract normal limit laws for the patterns when they are encoded, and perform asymptotic enumeration of the resulting classes when they are avoided. The results can be directly transferred in the case of labelled outerplanar graphs. We give examples and compute the relevant constants when the patterns are small cycles or dissections.Peer ReviewedPostprint (author's final draft
Encoding and avoiding 2-connected patterns in polygon dissections and outerplanar graphs
Let be a finite set of
2-connected patterns, i.e. graphs up to vertex relabelling. We study the
generating function which counts polygon
dissections and marks subgraph copies of with the variable . We
prove that this is always algebraic, through an explicit combinatorial
decomposition depending on . The decomposition also gives a defining
system for , which encodes polygon dissections that
restrict these patterns as subgraphs. In this way, we are able to extract
normal limit laws for the patterns when they are encoded, and perform
asymptotic enumeration of the resulting classes when they are avoided. The
results can be directly transferred in the case of labelled outerplanar graphs.
We give examples and compute the relevant constants when the patterns are small
cycles or dissections