Random subgraphs of certain graph powers

Random Subgraphs of Certain Graph Powers

Regular induced subgraphs of a random Graph

An old problem of Erdős, Fajtlowicz and Staton asks for the order of a largest induced regular subgraph that can be found in every graph on n vertices. Motivated by this problem, we consider the order of such a subgraph in a typical graph on n vertices, i.e., in a binomial random graph G(n, 1/2). We prove that with high probability a largest induced regular subgraph of G(n, 1/2) has about n ver...

Random Subgraphs in Cartesian Powers of Regular Graphs

Let G be a connected d-regular graph with k vertices. We investigate the behaviour of a spanning random subgraph Gp of G n, the n-th Cartesian power ofG, which is constructed by deleting each edge independently with probability 1−p. We prove that lim n→∞ P[Gp is connected] = e−λ, if p = p(n) = 1− ( λ 1/n n k )1/d and λn → λ > 0 as n → ∞. This extends a result of L. Clark, Random subgraphs of ce...

Critical Subgraphs of a Random Graph

We prove that the threshold for a random graph to have a k-core is equal to the threshold for having a subgraph which meets a necessary condition of Gallai for being k-critical. 1991 Mathematics Subject Classification: Primary 05C80; Secondary 05C15.

Domination number of graph fractional powers

For any $k in mathbb{N}$, the $k$-subdivision of graph $G$ is a simple graph $G^{frac{1}{k}}$, which is constructed by replacing each edge of $G$ with a path of length $k$. In [Moharram N. Iradmusa, On colorings of graph fractional powers, Discrete Math., (310) 2010, No. 10-11, 1551-1556] the $m$th power of the $n$-subdivision of $G$ has been introduced as a fractional power of $G$, denoted by ...