Long properly colored cycles in edge colored complete graphs

Abstract

Let K c n denote a complete graph on n vertices whose edges are colored in an arbitrary way. Let

Color degree and monochromatic degree conditions for short properly colored cycles in edge‐colored graphs

“…In the study of PC Hamilton cycles and long cycles in edge‐colored complete graphs, maximum monochromatic degree conditions are often involved (see and and the articles cited therein). Bollobás and Erdős conjectured that every edge‐colored complete graph with contains a PC Hamilton cycle.…”

Properly colored cycles in edge-colored graphs

“…Later, in [46], Lo showed the existence of a PC 2-factor under the condition of the above conjecture, and also verified the conjecture asymptotically. For other conjectures and results on PC Hamilton paths and cycles, we refer the reader to [1,2,4,13,20,22,24,26,45,46,53,62]. …”

“…Very recently, Lo [47] confirmed this conjecture asymptotically. When investigating long PC cycles, Wang et al [62] …”

“…So we start with the following important structural result: There are lots of results and problems on the existence of PC Hamilton cycles and long cycles (See [5,20,28,45,46,53,62]). For short PC cycles, especially, a PC triangle (or a rainbow triangle), the well-known Gallai coloring theory gives a structural characterization of edge-colored complete graphs containing no rainbow triangles (See [32] and [34]).…”

“…In the study of PC Hamilton cycles and long cycles in edge-colored complete graphs, maximum monochromatic degree conditions are often involved (See [47] and [62] and the papers cited therein). Bollobás and Erdős [15] conjectured that every edge-colored complete graph K n with…”