Hongtao Zhao

Let $G$ be a finite graph, $V(G)$ and $E(G)$ be its vertex set and edge set. A 1-factor $F$ of $G$ is a subgraph of $G$, which forms a matching of $V(G)$. A $m$-$cycle$ is a subgraph of $G$ with $m$ vertices $x_1,x_2,\ldots,x_m$ and $m$ edges $\{x_1,x_2\},\ldots,\{x_{m-1},x_m\},\{x_m,x_1\}$, which is denoted by ($x_1,x_2,\ldots,x_m$). Especially, when $m=|V(G)|$, the $m$-cycle is called \emph{Hamilton cycle} of $G$. A $m$-\emph{cycle decomposition} of $G$ is a partition of $E(G)$ into $m$-cycles. Obviously, there exists a $m$-cycle decomposition of $G$ only if the degree of each vertex in $G$ is even. But, for the graph $G$ in which the degree of each vertex is odd, we can consider the existence of $m$-cycle decomposition of $G-F$, where $F$ is a 1-factor of $G$. It is well known that there exist the following Hamilton cycle decompositions: $K_v$ for odd $v\geq3$, $K_v-F$ for even $v\geq4$, $K_{v,v}$ for even $v\geq2$, $K_{v,v}-F$ for odd $v\geq3$, where $K_v$ is the complete graph of order $v$ and $K_{v,v}$ is the complete bipartite graph with $v$ vertices in each partite set. A \emph{large set} of $m$-cycle decomposition of $G$ (resp. $G-F$) is a partition of all $m$-cycles of $G$ into $m$-cycle decompositions of $G$ (resp. $G-F$). There are many results regarding the existence of large sets of $m$-cycle decompositions. The existence of large sets of $3$-cycle decompositions of $K_{v}$, i.e., \emph{large sets of Steiner triple systems $LSTS(v)$}, has been completely solved by Jiaxi Lu (J. Combin. Theory Ser. A 34 (1983), 37 (1984)) and L. Teirlinck (J. Combin. Theory Ser. A 57 (1991)). In addition, there are some results for large sets of 4-cycle decompositions of $K_v$. The existence of large sets of $3$-cycle decompositions of $K_{v,v,\ldots,v}$, i.e., \emph{large sets of group divisible designs $LGDD(v^h)$}, has been completely solved by Jianguo Lei (J. Combin. Designs 5 (1997)). Darryn E. Bryant (Congr. Numer. 135 (1998)) has proved that there exist a large set of Hamilton cycle decomposition of $K_{2t+1}$ and a large set of Hamilton cycle decomposition of $K_{2t}- F$. In this paper, we will prove that there exist a large set of Hamilton cycle decomposition of $K_{2t,2t}$ and a large set of Hamilton cycle decomposition of $K_{2t+1,2t+1}- F$.