The generalized Mycielskians (also known as cones over graphs) are the natural generalization of the Mycielski graphs (which were first introduced by Mycielski in 1955). Given a graph G and any integer m 0, one can transform G into a new graph m(G), the generalized Mycielskian of G. This paper investigates circular clique number, total domination number, open packing number, fractional open pac...

Mycielski introduced a new graph transformation μ(G) for graph G, which is called the Mycielskian of G. A graph G is super connected or simply super-κ (resp. super edge connected or super-λ), if every minimum vertex cut (resp. minimum edge cut) isolates a vertex of G. In this paper, we show that for a connected graph G with |V (G)| ≥ 2, μ(G) is super-κ if and only if δ(G) < 2κ(G), and μ(G) is s...

Harary [10, p. 7] claims that Veblen [20, p. 2] first suggested to formalize simple graphs using simplicial complexes. We have developed basic terminology for simple graphs as at most 1-dimensional complexes. We formalize this new setting and then reprove Mycielski’s [12] construction resulting in a triangle-free graph with arbitrarily large chromatic number. A different formalization of simila...

The Mycielskian construction, denoted ?(G), takes a finite simple graph G to larger with of the same clique number but chromatic number. generalized ?t(G), odd girth. In this chapter we look at symmetry parameters ?(G) and ?t(G) in terms G. These include determining number, distinguishing cost distinguishing.

The packing chromatic number χρ(G) of a graphG is the smallest integer k such that the vertex set of G can be partitioned into sets Vi, i ∈ [k], where each Vi is an i-packing. In this paper, we investigate for a given triple (a, b, c) of positive integers whether there exists a graph G such that ω(G) = a, χ(G) = b, and χρ(G) = c. If so, we say that (a, b, c) is realizable. It is proved that b =...

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 ...