University of Isfahan Transactions on Combinatorics 2251-8657 14 4 2024 11 17 $G$-designs for the connected triangular bicyclic graphs with nine edges 271 281 28944 10.22108/toc.2024.140831.2156 EN Bryan Freyberg Department of Mathematics and Statistics, University of Minnesota Duluth, Duluth, MN Dalibor Froncek Department of Mathematics and Statistics University of Minnesota Duluth 1117 University Dr. Duluth, MN 55812-3000 USA 0000-0003-4528-2059 Joel Jeffries Department of Mathematics, Iowa State University, 411 Morrill Road, Ames, USA Gretta Jensen Department of Mathematics and Statistics, University of Minnesota Duluth, 1117 University Drive, Duluth, USA Andrew Sailstad School of Mathematics, University of Minnesota, 127 Vincent Hall 206 Church St. SE, Minneapolis, USA Journal Article 2024 02 27 A $G$-design of order $n$ is a decomposition of the complete graph $K_n$ into isomorphic copies of $G$. We show that if $G$ is a connected bicyclic graph with nine edges containing two triangles, a $G$-design of order $n$ exists whenever $n \equiv 0,1 \pmod{18}$. https://toc.ui.ac.ir/article_28944_0251192f0ac271abed0c719b103dc4d4.pdf