It is well known that a graph is bipartite if and only if it contains no odd cycles. Gallai characterized edge colorings of complete graphs containing no properly colored triangles in recursive sense. In this paper, we completely characterize edge-colored complete graphs containing no properly colored odd cycles and give an efficient algorithm with complexity \(O(n^{3})\) for deciding the existence of properly colored odd cycles in an edge-colored complete graph of order n. Moreover, we show that for two integers k, m with \(m\geqslant k\geqslant 3\), where \(k-1\) and m are relatively prime, an edge-colored complete graph contains a properly colored cycle of length \(\ell \equiv k\ (\text{mod}\ m)\) if and only if it contains a properly cycle of length \(\ell '\equiv k\ (\text{mod }\ m)\), where \(\ell '< 2m^{2}(k-1)+3m\).

众所周知，当且仅当它不包含奇数循环时，该图才是二分图。Gallai在递归意义上描述了完整图形的边缘着色，其中不包含正确着色的三角形。在本文中，我们完全刻画了不包含适当着色奇数周期的边缘着色完整图的特征，并给出了一种复杂度为\（O（n ^ {3}）\）的有效算法，用于确定边缘适当着色的奇周期n阶有色完整图。此外，我们表明对于两个整数k，  m具有\（m \ geqslant k \ geqslant 3 \），其中\（k-1 \）和m互质，边缘着色完整图包含长度的适当的彩色周期\（\ ELL \当量ķ\（\文本{模} \米）\）当且仅当它包含长度的适当周期\（\ ell'\ equiv k \（\ text {mod} \ m）\），其中\（\ ell'<2m ^ {2}（k-1）+ 3m \）。