| Summary: | 碩士 === 國立嘉義大學 === 資訊工程學系研究所 === 103 === Most of the graph coloring problems deal with vertices. This thesis discusses a kind of edge coloring that adjacent edges may be colored the same. A path P in graph G is a rainbow path if none of the edges in P are colored the same. The minimum number of colors used such that there is a rainbow path between each pair of distinct vertices, is the rainbow connection number of G, denoted as rc(G).
If it is desired that for each pair of vertices, there is a shortest path which is a rainbow path, then the least number of colors used to fulfill the condition is the strong rainbow connection number of G, denoted as src(G). It is known that for any connected graph G, the value of rc(G) and src(G) must lie between graph diameter and graph size (number of edges). This thesis investigated the sufficient and necessary conditions for graphs with (strong) rainbow connection number approaching to possible min/max value.
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