Proper vertex-pancyclicity of edge-colored complete graphs without joint monochromatic triangles

In an edge-colored graph (G,c), let dc(v) denote the number of colors on edges incident with a vertex v G and δc(G) minimum value over all vertices v∈V(G). A cycle (G,c) is called proper if any two adjacent have distinct colors. An n≥3 properly vertex-pancyclic each contained in length ℓ for every 3≤ℓ≤n. Fujita Magnant conjectured that complete δc(G)≥n+12 vertex-pancyclic. Chen, Huang Yuan partially solve this conjecture by adding extra condition does not contain monochromatic triangle. paper, we show true no joint triangles.