Research On Properly Colored Cycles In Edge-colored Complete Graphs With Forbidden Configurations

Given an edge-colored graph Gc,a subgraph of Gc is called properly colored if every two adjacent edges of this subgraph have distinct colors.A subgraph of Gc is called monochromatic if all the edges of this subgraph have the same color.The minimum color degree of Gc,denoted by ?c(Gc),is defined to be the maximum value of a nonnegative integer ? such that at least ? distinct colors appear on the edges incident with every vertex of Gc.A triangle refers a cycle of length 3.Lo(2013)showed that:If ?c(Gc)? 2,then G? contains a properly colored path of length 2?c(Gc)or a properly colored cycle of length ?c(Gc)+1.In this thesis,on the basis of the work of Lo(2013),we study the properly colored cycles in the edge-colored completed graphs Knc.According to the two cases where either the monochromatic triangle or monochromatic P4 is used as the forbidden configuration of Knc,we obtain the following results,respectively:(1)Let l1 and l2 be the lengthes of the shortest properly colored cycle and the longest properly colored cycle in Knc,respectively.If Knc contains no monochromatic triangles,then Knc contains a properly colored cycle of length l for any integer l with l1?l?l2.Furthermore,if ?c(Knc)=k? 2,then l1 ?4 and l2? min{2k-1,n}.(2)If Kc contains no monochromatic P4 and ?c(Knc)=k?3,then either Knc is properly vertex-pancyclic if n=4 and each vertex of Knc is contained a properly colored l-cycle for any integer 4 ?l?n-1 if n? 5 or a proper subset of V(Knc)becomes a degenerate set of Knc.