Proper distinguishing colorings with few colors for graphs with girth at least 5 (Q1658779)

Summary: The distinguishing chromatic number, \(\chi_D(G)\), of a graph \(G\) is the smallest number of colors in a proper coloring, \(\varphi\), of \(G\), such that the only automorphism of \(G\) that preserves all colors of \(\varphi\) is the identity map. \textit{K. L. Collins} and \textit{A. N. Trenk} [Electron. J. Comb. 13, No. 1, Research paper R16, 19 p. (2006; Zbl 1081.05033)] conjectured that if \(G\) is connected with girth at least 5 and \(G\neq C_6\), then \(\chi_D(G)\leqslant \Delta+1\). We prove this conjecture.