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

A proper coloring of the vertices of a graph is called a star coloring if every two color classes induce a star forest. Star colorings are a strengthening of acyclic colorings, i.e., proper colorings in which every two color classes induce a forest. We show that every acyclic $k$-coloring can be refined to a star coloring with at most $(2k^2-k)$ colors. Similarly, we prove that planar graphs have star colorings with at most 20 colors and we exhibit a planar graph which requires 10 colors. We prove several other structural and topological results for star colorings, such as: cubic graphs are $7$-colorable, and planar graphs of girth at least $7$ are $9$-colorable. We provide a short proof of the result of Fertin, Raspaud, and Reed that graphs with tree-width $t$ can be star colored with ${t+2\choose2}$ colors, and we show that this is best possible.

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This paper presents some recent results on lower bounds for independence ratios of graphs of positive genus and shows that in a limiting sense these graphs have the same independence ratios as do planar graphs. This last result is obtained by an application of Menger's Theorem to show that every triangulation of a surface of positive genus has a short cycle which does not separate the graph and is noncontractible on that surface.