Irregular independence and irregular domination

Abstract

If A is an independent set of a graph G such that the vertices in A have 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 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 α ir (G) and γ ir (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 α ir (G) = 1, we determine those that are planar, and we determine those that are outerplanar.

Singular Ramsey and Turán numbers

“…Starting from here, Albertson [2] proved 1 that for n ≥ 6 in every 2-coloring of the edges of K n there is a monochromatic K 3 with two equal degrees. Inspired by this result several papers were written, see for example [3,5,6,8,11].…”

“…Further related notions are the so-called fair dominating sets (which actually are regular dominating sets, see Caro, Hansberg and Henning [10]), irregular independence number and irregular domination number (Borg, Caro and Fenech [8]), and the problem of monochromatic degree-monotone paths in 2-colorings of the edges of complete graphs (Caro,Yuster and Zarb [12]).…”