Properly colored short cycles in edge-colored graphs

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

“…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.…”

“…Čada, Kaneko, Ryjáček, and Yoshimoto in [5] provided a minimum color degree condition for the existence of rainbow $${C}_{4}$$'s in edge‐colored triangle‐free graphs. For the existence of rainbow $${C}_{4}$$'s in general edge‐colored graphs, Ding, Hu, Wang, and Yang in [6] gave a minimum color degree condition.…”

Properly colored subgraphs in edge-colored graphs

“…In this chapter, we obtain a necessary and sufficient condition for edge-colored complete balanced bipartite graphs containing no monochromatic paths of length three to be PC even vertex-pancyclic. Our result is the following generalization of a result of Häggkvist and Manoussakis on bipartite tournaments in [Combinatorica 9 (1989) [33][34][35][36][37][38]. If K c n,n contains no monochromatic paths of length three and K c n,n has a PC Hamilton cycle, then K c n,n is PC even vertex-pancyclic, unless K c n,n belongs to two special classes of edge-colored graphs.…”

“…Čada et al [25] in 2016 gave a minimum color degree condition for the existence of rainbow 4-cycles in triangle-free edge-colored graphs. Ding et al [33] in 2022 gave an asymptotically sharp color degree condition for the existence of rainbow 4-cycles in edge-colored graphs.…”

“…Independently, Čada et al [26] in 2020 characterized edge-colored complete bipartite graphs containing no PC 4-cycles. 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.…”