Graphs with Hamiltonian cycles having adjacent lines different colors

Hamiltonian problems in edge-colored complete graphs and eulerian cycles in edge-colored graphs : some complexity results

“…Results in almost the same vein are proved in [6]. Also, in [1], necessary and sufficient conditions are presented (see theorem 1 below).…”

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]. …”

“…Early research on the existence of PC Hamilton cycles in fact dates back to the 1970s [15,20,24,53], but this topic has also attracted new interest more recently [5,47]. Similarly, the existence of PC triangles has been studied by different research groups during the same period as well [27,32,34,38,39].…”

“…Similarly, the existence of PC triangles has been studied by different research groups during the same period as well [27,32,34,38,39]. These topics have been dealt with for general graphs [27,33,38,39,[44][45][46]65] but also for complete graphs [5,15,20,24,27,28,32,34,42,43,47,53] and for complete bipartite graphs [5,20,42]. Moreover, the theory involved in the study of PC cycles in edge-colored graph is closely related to the theory of directed cycles in directed graphs.…”

“…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]).…”