A labeling of the vertices of a graph G, $\phi :V(G) \rightarrow \{1,\ldots,r\}$, is said to be $r$-distinguishing provided no automorphism of the graph preserves all of the vertex labels. The distinguishing number of a graph G, denoted by $D(G)$, is the minimum $r$ such that $G$ has an $r$-distinguishing labeling. The distinguishing number of the complete graph on $t$ vertices is $t$. In contrast, we prove (i) given any group $\Gamma$, there is a graph $G$ such that $Aut(G) \cong \Gamma$ and $D(G)= 2$; (ii) $D(G) = O(log(|Aut(G)|))$; (iii) if $Aut(G)$ is abelian, then $D(G) \leq 2$; (iv) if $Aut(G)$ is dihedral, then $D(G) \leq 3$; and (v) If $Aut(G) \cong S_4$, then either $D(G) = 2$ or $D(G) = 4$. Mathematics Subject Classification 05C,20B,20F,68R
In this paper we define and study the distinguishing chromatic number, $\chi_D(G)$, of a graph $G$, building on the work of Albertson and Collins who studied the distinguishing number. We find $\chi_D(G)$ for various families of graphs and characterize those graphs with $\chi_D(G)$ $ = |V(G)|$, and those trees with the maximum chromatic distingushing number for trees. We prove analogs of Brooks' Theorem for both the distinguishing number and the distinguishing chromatic number, and for both trees and connected graphs. We conclude with some conjectures.
Hypoxia and TGF-beta1 treatments of cultured HPF modulate the expression of TGF-beta1, beta2 and beta3 and their receptors betaIR and betaIIR by increasing the ratio of TGF-beta1 and beta2 to beta3, thus favoring the development of peritoneal adhesion.
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