Proper Hamiltonian cycles in edge-colored multigraphs

Abstract

A c-edge-colored multigraph has each edge colored with one of the c available colors and no two parallel edges have the same color. A proper Hamiltonian cycle is a cycle containing all the vertices of the multigraph such that no two adjacent edges have the same color. In this work we establish sufficient conditions for a multigraph to have a proper Hamiltonian cycle, depending on several parameters such as the number of edges and the rainbow degree.

Compatible spanning circuits in edge-colored graphs

“…In this context, it is worth mentioning that Abouelaoualim et al [1] established a sufficient condition in terms of the minimum monochromatic degree for the existence of compatible Hamilton cycles, when considering the existence of properly colored cycles of all possible lengths in edge-colored multigraphs. More results and problems (conjectures) related to the existence of compatible Hamilton cycles in specific edge-colored graphs can be found in [1,2,4,23,24,54,74,92].…”

Properly colored cycles in edge-colored graphs

“…Let t be a smallest integer in [1, −1] such that col(uw t ) = α. Similar to the proof in the case that k ≥ 1, there must exist a vertex y 0 ∈ {w 1 , w 2 …”

“…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]. …”

Compatible spanning circuits in edge-colored graphs

“…We refer the reader to [25,28,29] for more results on the existence of compatible Hamilton cycles in 2-edge-colored (multi)graphs. The research on the existence of compatible Hamilton cycles in specific edge-colored graphs without restrictions on the number of colors dates back to the 1970s (see [12,23,31,111]), and this topic has also attracted new attention recently (see [1,2,4,92]). On the other hand, Kotzig [84]…”

“…InTheorem 4.3.3, it is worth noting that Condition (1) implies Condition(2), and Condition(5) implies Condition(4).…”