In the context of an edge-coloured graph , a path within the graph is deemed when a colour is exclusively applied to one of its edges. The presence of a conflict-free path connecting any two unique vertices of an edge-coloured graph is what defines it as . The , indicated by , is the fewest number of colours necessary to make conflict-free connected. Consider the subgraph of a connected graph , which is constructed from the cut-edges of . Let be the minimum degree-sum of any 3 independent vertices in . In this study, we establish that for a connected graph with an order of and , the following conditions hold: (1) when ; (2) when forms a linear forest. Moreover, we will now demonstrate that if is a connected, non-complete graph with vertices, where , is a linear forest, , and , then . Moreover, we also determine the upper bound of the number of cut-edges of a connected graph depending on the degree-sum of any three independent vertices.

中文翻译：

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图的无冲突连接数和最小度和


在边缘彩色图的上下文中，当颜色专门应用于其边缘之一时，就被认为是图中的路径。连接边彩色图的任意两个唯一顶点的无冲突路径的存在将其定义为 。由 表示的 是实现无冲突连接所需的最少颜色数。考虑连通图 的子图，它是由 的切边构造的。令 为 中任意 3 个独立顶点的最小度数和。在本研究中，我们建立了对于阶数为 和 的连通图，以下条件成立： (1) 当 ; (2)何时形成线性森林。此外，我们现在将证明 if 是一个有顶点的连通非完备图，其中 , 是一个线性森林， , 和 , then 。此外，我们还根据任意三个独立顶点的度和来确定连通图的切边数量的上限。