On the number of independent and $k$-dominating sets in graphs with average vertex degree at most $k$

Abstract

The following conjecture is formulated: if the average vertex degree in a graph is not greater than a positive integer $k \geqslant 1$, then the number of $k$-dominating sets in this graph does not exceed the number of its independent sets, and these numbers are equal to each other if and only if the graph is $k$-regular. This conjecture is proved for $k \in \{1,2\}$. Bibliography: 10 titles.