The main focus of this thesis is to find edge-disjoint multicolored subgraphs in a properly edge-colored complete graph. If the complete graph is properly k-edge-colored, then we are aiming to obtain edge-disjoint copies of multicolored subgraphs of size k. This is why we try to find copies of multicolored spanning trees of K2msince it is (2m −1)-edge-colorable and find copies of multicolored spanning unicyclic subgraphs of K2m+1 since it is (2m+1)-edge-colorable.
In case that the proper edge-coloring is of special type or prescribed, then in Chapter 2 and Chapter 3 we have an M T P (multicolored spanning tree parallelism) or an M HCP (multicolored Hamiltonian cycle parallelism) respectively when K2m or K2m+1 are con-sidered. However, if the proper edge-colorings are arbitrarily given, then finding copies of multicolored subgraph is going to be very difficult. In fact, except for special graphs such as stars, small paths or small cycles, finding just one copy (multicolored) of a given graph, for example, a multicolored perfect matching in K2m, is difficult enough.
Therefore, we put our effort in searching for edge-disjoint (not necessarily be iso-morphic) multicolored spanning trees in a properly (2m−1)-edge-colored K2m and mul-ticolored unicyclic spanning subgraphs in a properly (2m+1)-edge-colored K2m+1 respec-tively. In Chapter 4 and Chapter 5, by using a recursive construction, we are able to find Ω(√
m) edge-disjoint multicolored spanning trees and Ω(√
m) edge-disjoint multicolored spanning unicylic subgraphs in K2m and K2m+1 respectively. Though this result is the
best one obtained so far, it is very far from m spanning trees (conjectured by Brauldi and Hollingsworth) and m unicyclic spanning subgraphs (conjectured by Constantine).
Hopefully, we can close the gap in the near future.
In this thesis, we also consider ”forbidden” multicolored subgraphs in a properly edge-colored complete bipartite graph. Mainly, we prove that if the two partite sets are large enough, then forbidding a multicolored even cycle of fixed length is not possible. Pre-cisely, we prove that for f (k) ≤ n, then every properly n-edge-colored Kk,n contains a multicolored 2k-cycle where f (k) = 5k− 6. As a consequence, we determine the set of all ordered pairs (m, n), such that multicolored C6 can be forbidden in Km,n. Unfortunately, determining the set of (m, n)’s such that multicolored C2k can be forbidden in Km,n (by giving a proper n-edge-coloring) is still unsolved. We believe that it is close related to find a latin rectangle with special structure which is worth of more study.
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