Properly colored C4’s in edge-colored graphs

Properly colored and rainbow C4 ${C}_{4}$'s in edge‐colored graphs

“…For the existence of PC $${C}_{4}$$'s in edge‐colored complete graphs, Axenovich, Jiang, and Tuza in [3] gave a sharp sufficient minimum color degree condition. More recently, Xu, Magnant, and Zhang in [17] provided a sharp sufficient color number condition, and very recently Han, Broersma, Bai, and Zhang in [11] gave a sharp sufficient maximum monochromatic degree condition. A few years ago, Fujita, Li, and Zhang in [8] characterized the structure of edge‐colored complete bipartite graphs containing no PC $${C}_{4}$$'s, and gave minimum color degree and maximum monochromatic degree conditions for an edge‐colored complete bipartite graph to contain a PC $${C}_{4}$$.…”

“…A few years ago, Fujita, Li, and Zhang in [8] characterized the structure of edge‐colored complete bipartite graphs containing no PC $${C}_{4}$$'s, and gave minimum color degree and maximum monochromatic degree conditions for an edge‐colored complete bipartite graph to contain a PC $${C}_{4}$$. For the existence of PC $${C}_{4}$$'s in general edge‐colored graphs, Xu, Magnant, and Zhang in [17] provided a sharp sufficient condition in terms of the number of edges plus the number of colors, and a sharp color degree sum condition, whereas Ding, Hu, Wang, and Yang in [6] this year established a sufficient minimum color degree condition.…”

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

“…Recently, Magnant et al [8] studied the existence of monochromatic cliques, cycles, and stars in colored complete graphs that contain no PC . Xu et al [10] determined the structure of an ‐colored containing no PC and gave sufficient conditions for the existence of PC 's in edge‐colored graphs. From a computational complexity angle, Gutin et al [7] studied the complexity of determining the existence of odd PC cycles in edge‐colored graphs.…”

Properly colored subgraphs in edge-colored graphs

“…Ding et al [33] in 2022 obtained an asymptotically sharp color degree condition for the existence of PC 4-cycles and rainbow 4-cycles in edgecolored graphs. For more related work, see [2,35,36,57] for the existence of PC triangles and [25,44,89] for PC (or rainbow, i.e., with all edges colored differently) 4-cycles, respectively.…”