Decomposition of a \(2K_{10t+5}\) into \(H_{3}\) Graphs

An \( H_3 \) graph is a multigraph on three vertices with double edges between two pairs of distinct vertices and a single edge between the third pair. To settle the \( H_3 \) decomposition problem completely, one needs to complete the decomposition of a \( 2K_{10t+5} \) into \( H_3 \) graphs. In this paper, we present two new construction methods for such decompositions, resulting in previously unknown decompositions for \( v = 15, 25, 35, 45 \) and two new infinite families.