For an edge-colored graph, its minimum color degree is defined as the minimum number of colors appearing on the edges incident to a vertex and its maximum monochromatic degree is defined as the maximum number of edges incident to a vertex with a same color. A cycle is called properly colored if every two of its adjacent edges have distinct colors. In this article, we first give a minimum color degree condition for the existence of properly colored cycles, then obtain the minimum color degree condition for an edge-colored complete graph to contain properly colored triangles. Afterwards, we characterize the structure of an edge-colored complete bipartite graph without containing properly colored cycles of length 4 and give the minimum color degree and maximum monochromatic degree conditions for an edge-colored complete bipartite graph to contain properly colored cycles of length 4, and those passing through a given vertex or edge, respectively.

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

Rainbow cycles in edge-colored graphs

Let G be an edge-colored graph and v a vertex of G. The color degree of v is the number of colors appearing on the edges incident to v. A rainbow triangle in G is one in which all edges have distinct colors. In this paper, we first prove that an edge-colored graph on n vertices contains a rainbow triangle if the color degree sum of every two adjacent vertices is at least $$n+1$$n+1. Afterwards, we characterize the edge-colored graphs on n vertices containing no rainbow triangles but satisfying that each pair of adjacent vertices has color degree sum at least n.

Color Degree Sum Conditions for Rainbow Triangles in Edge-Colored Graphs

With respect to specific cycle-related problems, edge-colored graphs can be considered as a generalization of directed graphs. We show that properly edge-colored theta graphs play a key role in characterizing the difference between edge-colored complete graphs and multipartite tournaments. We also establish sufficient conditions for an edge-colored complete graph to contain a small and a large properly edge-colored theta graph, respectively.

/pdf/properly-edge-colored-theta-graphs-in-edge-colored-complete-25chsechuj.pdf

Properly Edge-colored Theta Graphs in Edge-colored Complete Graphs

Rainbow triangles and cliques in edge-colored graphs

Rainbow C3’s and C4’s in edge-colored graphs

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Rainbow C3’s and C4’s in edge-colored graphs