在本論文中，我們研究「邊著色圖」上的「有色約束生成樹問題」和「出度有色約束生成樹問題」。這些問題旨在尋找一個生成樹，其中每個節點的任意特定顏色的鄰邊（或出邊），其邊數不超過一個給定的常數。我們證明了當輸入的圖為有向圖時，這兩個問題皆為NP困難。此外，即使輸入為有向無環圖時，有色約束生成樹問題仍然是NP困難的。

In this thesis, we study the Colored Constrained Spanning Tree Problem (CCST problem) and the Colored Out-Constrained Spanning Tree Problem (COCST problem) on edge-colored graphs.
These problems aim to find a spanning tree such that for each vertex, the number of incident edges (or outgoing edges) sharing any specific color is bounded by a constant.
We demonstrate the NP-hardness of both problems when the input graphs are directed graphs.
Additionally, even when considering directed acyclic graphs (DAGs) as input, the CCST problem remains NP-hard.

Abstract (Chinese) i
Abstract ii
Acknowledgment iii
Contents v
1 Introduction 1
1.1 Thesis Organization . . . . . . . . . . . . . . . 2
1.2 Related Work . . . . . . . . . . . . . . . . . . . 2
1.3 Definitions . . . . . . . . . . . . . . . . . . . 3
1.4 Our Contributions . . . . . . . . . . . . . . . . 4
2 The κ-CCST Problem on DAGs 6
2.1 Proof of Claim 1 . . . . . . . . . . . . . . . . . 11
2.2 Proof of Claim 2 . . . . . . . . . . . . . . . . . 12
2.3 Extending to Bipartite Graphs . . . . . . . . . . 14
3 The κ-COCST Problem on Directed Graphs 15
3.1 Reducing κ-CCST to κ-COCST . . . . . . . . . . . 16
3.2 Proof of Claim 3 . . . . . . . . . . . . . . . . . 18
3.3 Proof of Claim 4 . . . . . . . . . . . . . . . . . 20
4 Conclusion 23

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