Combinatorics of $k$-Farey graphs

k-Efficient partitions of graphs

A set $S = {u_1,u_2, ldots, u_t}$ of vertices of $G$ is an efficientdominating set if every vertex of $G$ is dominated exactly once by thevertices of $S$. Letting $U_i$ denote the set of vertices dominated by $u_i$%, we note that ${U_1, U_2, ldots U_t}$ is a partition of the vertex setof $G$ and that each $U_i$ contains the vertex $u_i$ and all the vertices atdistance~1 from it in $G$. In this ...

Farey graphs as models for complex networks

Farey sequences of irreducible fractions between 0 and 1 can be related to graph constructions known as Farey graphs. These graphs were first introduced by Matula and Kornerup in 1979 and further studied by Colbourn in 1982 and they have many interesting properties: they are minimally 3-colorable, uniquely Hamiltonian, maximally outerplanar and perfect. In this paper we introduce a simple gener...

k-Remainder Cordial Graphs

In this paper we generalize the remainder cordial labeling, called $k$-remainder cordial labeling and investigate the $4$-remainder cordial labeling behavior of certain graphs.

Analytic Combinatorics of Random Graphs

Analytic combinatorics of connected graphs

We enumerate the connected graphs that contain a linear number of edges with respect to the number of vertices. So far, only first term of the asymptotics and a bound on the error were known. Using analytic combinatorics, i.e. generating function manipulations, we derive the complete asymptotic expansion. keywords. connected graphs, analytic combinatorics, generating functions, asymptotic expan...