Edge‐colored complete graphs without properly colored even cycles: A full characterization

Abstract

The structure of edge-colored complete graphs containing no properly colored triangles has been characterized by Gallai back in the 1960s. More recently, Cǎda et al. and Fujita et al. independently determined the structure of edge-colored complete bipartite graphs containing no properly colored C 4 . We characterize the structure of edge-colored complete graphs containing no properly colored even cycles, or equivalently, without a properly colored C 4 or C 6 . In particular, we first deal with the simple case of 2-edge-colored complete graphs, using a result of Yeo. Next, for ≥ k 3, we define four classes of k-edgecolored complete graphs without properly colored even cycles and prove that any k-edge-colored complete graph without a properly colored even cycle belongs to one of these four classes.This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.

A revisit to Bang-Jensen-Gutin conjecture and Yeo's theorem

“…Furthermore, the bound is best possible. Besides, Yeo's theorem is frequently used not only on the existence of PC cycles (see the survey paper [1] and recent results [8,13,17,19]) but also on other topics, such as PC trees [7], decomposition of edge-colored graphs [12],…”

Properly colored subgraphs in edge-colored graphs

“…Finally, we focus on PC even cycles and PC odd cycles in edge-colored complete graphs. R. Li et al [67] characterized edge-colored complete graphs containing no PC even cycles, which is equivalent to characterizing edgecolored bipartite complete graphs containing no PC 4-cycles and PC 6-cycles. In Chapter 2, we characterize edge-colored complete graphs containing no PC odd cycles, which is equivalent to characterizing edge-colored complete graphs containing no PC triangles and PC 5-cycles.…”

“…We postpone the proof of this theorem to Section 2.3. Moreover, Li et al [67] We postpone the proof of this theorem to Section 2.4. In next section, we present some preliminaries.…”