($4$, $5)$-CYCLE SYSTEMS OF COMPLETE MULTIPARTITE GRAPHS

Abstract

In 1981, Alspach conjectured that if $3 \leq m_{i} \leq v$, $v$ is odd and $v(v-1)/2 = m_{1} + m_{2} + \dots + m_{t}$, then the complete graph $K_{v}$ can be decomposed into $t$ cycles of lengths $m_{1}$, $m_{2}$, $\dots$, $m_{t}$ respectively; if $v$ is even, $v(v-2)/2 = m_{1} + m_{2} + \dots + m_{t}$, then the complete graph minus a one-factor $K_{v}-F$ can be decomposed into $t$ cycles of lengths $m_{1}$, $m_{2}$, $\dots$, $m_{t}$ respectively. In this paper, we extend the study to the decomposition of the complete equipartite graph $K_{m(n)}$. For $m_{i} \in \{4,5\}$, we prove that the trivial necessary conditions are also sufficient.