Irregular independence and irregular domination

If A is an independent set of a graph G such that the vertices in A have pairwise different degrees, then we call A an irregular independent set of G. If D is a dominating set of G such that the vertices that are not in D have pairwise different numbers of neighbours in D, then we call D an irregular dominating set of G. The size of a largest irregular independent set of G and the size of a smallest irregular dominating set of G are denoted by αir(G) andγir(G), respectively. We initiate the investigation of these two graph parameters. For each of them, we obtain sharp bounds in terms of basic graph parameters such as the order, the size, the minimum degree and the maximum degree, and we obtain Nordhaus–Gaddum-type bounds. We also establish sharp bounds relating the two parameters. Furthermore, we characterize the graphs G with αir(G) = 1, we determine those that are planar, and we determine those that are outerplanar.peer-reviewe

Singular Ramsey and Turán numbers

We say that a subgraph F of a graph G is singular if the degrees dG(v) are all equal or all distinct for the vertices v ∈ V (F). The singular Ramsey number Rs(F) is the smallest positive integer n such that, for every m at least n, in every edge 2-coloring of Km, at least one of the color classes contains F as a singular subgraph. In a similar flavor, the singular Turán number Ts(n,F) is defined as the maximum number of edges in a graph of order n, which does not contain F as a singular subgraph. In this paper we initiate the study of these extremal problems. We develop methods to estimate Rs(F) and Ts(n,F), present tight asymptotic bounds and exact results