Let G be an edge-colored connected graph. A path P is a proper path in G if no two adjacent edges of P are colored the same. If P is a proper u − v path of length d(u, v), then P is a proper u − v geodesic. An edge coloring c is a proper-path coloring of a connected graph G if every pair u, v of distinct vertices of G are connected by a proper u − v path in G, and c is a strong proper-path coloring if every two vertices u and v are connected by a proper u − v geodesic in G. The minimum number of colors required for a proper-path coloring or strong proper-path coloring of G is called the proper connection number pc(G) or strong proper connection number spc(G) of G, respectively. If G is a nontrivial connected graph of size m, then pc(G) ≤ spc(G) ≤ m and pc(G) = m or spc(G) = m if and only if G is the star of size m. In this paper, we determine all connected graphs G of size m for which pc(G) or spc(G) is m − 1, m − 2 or m − 3.

In the noisy channel model from coding theory, we wish to detect errors introduced during transmission by optimizing various parameters of the code. Bennett, Dudek, and LaForge framed a variation of this problem in the language of alternating paths in edge-colored complete bipartite graphs in 2016. Here, we extend this problem to the random graph G(n, p). We seek the alternating connectivity, κ r,ℓ (G), which is the maximum t such that there is an r-edge-coloring of G such that any pair of vertices is connected by t internally disjoint and alternating (i.e. no consecutive edges of the same color) paths of length ℓ. We have three main results about how this parameter behaves in G(n, p) that basically cover all ranges of p: one for paths of length two, one for the dense case, and one for the sparse case. For paths of length two, we found that κ r,ℓ (G) is essentially the codegree of a pair of vertices. For the dense case when p is constant, we were able to achieve the natural upper bounds of minimum degree (minus some intersection) or the total number of disjoint paths between a pair of vertices. For the sparse case, we were able to find colorings that achieved the natural obstructions of minimum degree or (in a slightly less precise result) the total number of paths of a certain length in a graph. We broke up this sparse case into ranges of p corresponding to when G(n, p) has diameter k or k + 1. We close with some remarks about a similar parameter and a generalization to pseudorandom graphs.

No abstract