Edge-Colored Complete Graphs Containing No Properly Colored Odd Cycles

Abstract

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 $\ell '< 2m^{2}(k-1)+3m$.