Proper Cycles and Rainbow Cycles in 2-triangle-free edge-colored Complete Graphs

An edge-colored graph is called rainbow if every two edges receive distinct colors, and called proper if every two adjacent edges receive distinct colors. Anedge-colored graph G(c) is properly vertex/edge-pancyclic if every vertex/edge of the graph is contained in a proper cycle of length k for every k with 3 <= k <= vertical bar V(G)vertical bar. A triangle C of G(c) is called a 2-triangle if C receives exactly two colors. We call G(c) 2-triangle-free if G(c) contains no 2-triangle. Let d(c)(v) be the number of colors on the edges incident to v in G(c) and let delta(c)(G) be the minimum d(c)(v) over all the vertices v is an element of V(G(c)). We show in this paper that: (i) a 2-triangle-free edge-colored complete graph K-n(c) is properly vertex-pancyclic if delta(c)(K-n) >= inverted right perpendicularn/3inverted left perpendicular + 1, and is properly edge-pancyclic if delta(c)(K-n) >= inverted right perpendicularn/3inverted left perpendicular + 2. (ii) with the exception of a few edge-colored graphs on at most 9 vertices, every vertex of a 2-triangle-free edge-colored complete graph K-n(c) with delta(c)(K-n) >= 4 is contained in a rainbow C4; (iii) every vertex of a 2-triangle-free edge-colored complete graph K-n(c) with delta(c)(K-n) >= 5 is contained in a rainbow C-5 unless G is proper or G is a special edge-colored K-8.