The distinguishing index of graphs with infinite minimum degree

The distinguishing index D & PRIME;(G) $D<^>{\prime} (G)$ of a graph G $G$ is the least number of colors necessary to obtain an edge coloring of G $G$ that is preserved only by the trivial automorphism. We show that if G $G$ is a connected & alpha; $\alpha $-regular graph for some infinite cardinal & alpha; $\alpha $ then D & PRIME;(G)& LE;2 $D<^>{\prime} (G)\le 2$, proving a conjecture of Lehner, Pilsniak, and Stawiski. We also show that if G $G$ is a graph with infinite minimum degree and at most 2 & alpha; ${2}<^>{\alpha }$ vertices of degree & alpha; $\alpha $ for every infinite cardinal & alpha; $\alpha $, then D & PRIME;(G)& LE;3 $D<^>{\prime} (G)\le 3$. In particular, D & PRIME;(G)& LE;3 $D<^>{\prime} (G)\le 3$ if G $G$ has infinite minimum degree and order at most 2 & ALEPH;0 ${2}<^>{{\aleph }_{0}}$.