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Register a new userEnumeration of labelled 4-regular planar graphs
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We present the first combinatorial scheme for counting labelled 4-regular planar graphs through a complete recursive decomposition. More precisely, we show that the exponential generating function of labelled 4-regular planar graphs can be computed effectively as the solution of a system of equations, from which the coefficients can be extracted. As a byproduct, we also enumerate labelled 3-connected 4-regular planar graphs, and simple 4-regular rooted maps.
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Building on previous work by the present authors [Proc. London Math. Soc. 119(2):358--378, 2019], we obtain a precise asymptotic estimate for the number $g_n$ of labelled 4-regular planar graphs. Our estimate is of the form $g_n sim gcdot n^{-7/2} rho^{-n} n!$, where $g>0$ is a constant and $rho approx 0.24377$ is the radius of convergence of the generating function $sum_{nge 0}g_n x^n/n!$, and conforms to the universal pattern obtained previously in the enumeration of planar graphs. In addition to analytic methods, our solution needs intensive use of computer algebra in order to work with large systems of polynomials equations. In particular, we use evaluation and Lagrange interpolation in order to compute resultants of multivariate polynomials. We also obtain asymptotic estimates for the number of 2- and 3-connected 4-regular planar graphs, and for the number of 4-regular simple maps, both connected and 2-connected.
Point-determining graphs are graphs in which no two vertices have the same neighborhoods, co-point-determining graphs are those whose complements are point-determining, and bi-point-determining graphs are those both point-determining and co-point-determining. Bicolored point-determining graphs are point-determining graphs whose vertices are properly colored with white and black. We use the combinatorial theory of species to enumerate these graphs as well as the connected cases.
In this paper we are interested in the asymptotic enumeration of Cayley graphs. It has previously been shown that almost every Cayley digraph has the smallest possible automorphism group: that is, it is a digraphical regular representation (DRR). In this paper, we approach the corresponding question for undirected Cayley graphs. The situation is complicated by the fact that there are two infinite families of groups that do not admit any graphical regular representation (GRR). The strategy for digraphs involved analysing separately the cases where the regular group $R$ has a nontrivial proper normal subgroup $N$ with the property that the automorphism group of the digraph fixes each $N$-coset setwise, and the cases where it does not. In this paper, we deal with undirected graphs in the case where the regular group has such a nontrivial proper normal subgroup.
Szemeredis Regularity Lemma is a very useful tool of extremal combinatorics. Recently, several refinements of this seminal result were obtained for special, more structured classes of graphs. We survey these results in their rich combinatorial context. In particular, we stress the link to the theory of (structural) sparsity, which leads to alternative proofs, refinements and solutions of open problems. It is interesting to note that many of these classes present challenging problems. Nevertheless, from the point of view of regularity lemma type statements, they appear as gentle classes.
In this article, we discuss when one can extend an r-regular graph to an r + 1 regular by adding edges. Different conditions on the num- ber of vertices n and regularity r are developed. We derive an upper bound of r, depending on n, for which, every regular graph G(n, r) can be extended to an r + 1-regular graph with n vertices. Presence of induced complete bipartite subgraph and complete subgraph is dis- cussed, separately, for the extension of regularity.
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