Research On Proper Vertex-Pancyclicity Of Edge-Colored Complete Graphs Without Monochromatic Triangles

In an edge-colored graph(G,C),let dc(v)be the number of colors on the edges incident to vertex v and let ?c(G)be the minimum value of dc(v)over all vertices v ? 17(G).Consider a subgraph in an edge-colored graph(G,c).The subgraph is called proper if every two adjacent edges in the subgraph are colored by distinct colors.The subgraph is called rainbow if all its edges in the subgraph are colored by distinct colors.The subgraph is called monochromatic if all its edges in the subgraph are colored by the same color.In an edge-colored graph(G,c),let C= v1v2...vlv1 be a cycle and v be a vertex in V(G)\V(C)and let vi+1=v1 and v0=vl.We say that v is a monochromatic vertex of C if the edges in(?)(v,V(C))have just one color in(G,c).We say that v follows the colors of C increasingly if c(vvi)= c(vvi+1)for all i=1,2,...,l,and v follows the colors of C decreasingly if c(vvi)=c(viv-1)for all i=1,2,...,l.In either cases.we say that the vertex v is a decreasing(increasing)following vert:x of C.An edge-colored graph(G,c)on n?3 vertices is called properly vertex-pancyclic if each vertex of(G,c)is contained in a proper cycle of length l for every l with 3?l?n.In 2011,Fujita and Magnant conjectured that,every edge-colored complete graph on n?3 vertices with ?c(G)?n+1/2 is properly vert,ex-pancyclic.In this thesis,we show that the conjecture holds uncder some specific conditions.The main results of this thesis are as follows:(1)Let(G,c)be an edge-colored complete graph on n?3 vertices such that ?c(G)?n+1/2.If(G,c)contains no rmonochromatic triangles,then(G,c)is properly vertex-pancyclic.(2)Let(G,c)be an edge-colored complete graph on n?3 vertices such that?c(G)?n+1/2.If(G,c)contains no intersecting monochromatic triangles,and there are monochromatic vertices or at least two following vertices for any proper cycle of(G,c),then(G,c)is properly vertex-pancyclic.