Given two disjoint copies of a graph $G$, denoted ${G}^{1}$ and ${G}^{2}$, and a permutation $\pi $ of $V\left(G\right)$, the graph $\pi G$ is constructed by joining $u\in V\left({G}^{1}\right)$ to $\pi \left(u\right)\in V\left({G}^{2}\right)$ for all $u\in V\left({G}^{1}\right)$. $G$ is said to be a universal fixer if the domination number of $\pi G$ is equal to the domination number of $G$ for all $\pi $ of $V\left(G\right)$. In 1999 it was conjectured that the only universal fixers are the edgeless graphs. Since then, a few partial results have been shown. In this paper, we prove the conjecture completely.

Given a coloring of the vertices, we say subgraph H is monochromatic if every vertex of H is assigned the same color, and rainbow if no pair of vertices of H are assigned the same color. Given a graph G and a graph F, we define an F-WORM coloring of G as a coloring of the vertices of G without a rainbow or monochromatic subgraph H isomorphic to F. We present some results on this concept especially as regards to the existence, complexity, and optimization within certain graph classes. The focus is...

A variation of graph coloring known as a t-tone k-coloring assigns a set of t colors to each vertex of a graph from the set {1, . . . , k}, where the sets of colors assigned to any two vertices distance d apart share fewer than d colors in common. The minimum integer k such that a graph G has a t- tone k-coloring is known as the t-tone chromatic number. We study the 2-tone chromatic number in three different graph products. In particular, given graphs G and H, we bound the 2-tone chromatic number...

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