We study a new variant of connected coloring of graphs based on the concept of strong edge coloring (every color class forms an induced matching). In particular, an edge-colored path is strongly proper if its color sequence does not contain identical terms within a distance of at most two. A strong proper connected coloring of G is the one in which every pair of vertices is joined by at least one strongly proper path. Let \({{\,\textrm{spc}\,}}(G)\) denote the least number of colors needed for such coloring of a graph G. We prove that the upper bound \({{\,\textrm{spc}\,}}(G)\le 5\) holds for any 2-connected graph G. On the other hand, we demonstrate that there are 2-connected graphs with arbitrarily large girth satisfying \({{\,\textrm{spc}\,}}(G)\ge 4\). Additionally, we prove that graphs whose cycle lengths are divisible by 3 satisfy \({{\,\textrm{spc}\,}}(G)\le 3\). We also consider briefly other connected colorings defined by various restrictions on color sequences of connecting paths. For instance, in a nonrepetitive connected coloring of G, every pair of vertices should be joined by a path whose color sequence is nonrepetitive, that is, it does not contain two adjacent identical blocks. We demonstrate that 2-connected graphs are 15-colorable, while 4-connected graphs are 6-colorable, in the connected nonrepetitive sense. A similar conclusion with a finite upper bound on the number of colors holds for a much wider variety of connected colorings corresponding to fairly general properties of sequences. We end the paper with some open problems of concrete and general nature.

我们基于强边缘着色的概念研究了图的连接着色的新变体（每个颜色类别形成诱导匹配）。特别是，如果边缘颜色路径的颜色序列在最多两个距离内不包含相同的项，则该路径是强正确的。 G的强真连通着色是每对顶点都由至少一条强真路径连接的着色。让\({{\,\textrm{spc}\,}}(G)\)表示图G的这种着色所需的最少颜色数。我们证明上界\({{\,\textrm{spc}\,}}(G)\le 5\)对于任何 2-连通图G都成立。另一方面，我们证明存在任意大周长的 2 连通图满足\({{\,\textrm{spc}\,}}(G)\ge 4\)。此外，我们证明循环长度可被 3 整除的图满足\({{\,\textrm{spc}\,}}(G)\le 3\)。我们还简要考虑了由连接路径颜色序列的各种限制定义的其他连接着色。例如，在G的非重复连接着色中，每对顶点都应该由一条颜色序列为非重复的路径连接，即不包含两个相邻的相同块。我们证明，在连通非重复意义上，2 连通图具有 15 种颜色，而 4 连通图具有 6 种颜色。颜色数量有限上限的类似结论适用于与序列的相当一般属性相对应的更广泛的连接颜色。我们以一些具体和一般性的开放性问题来结束本文。