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Proper Connection Of Direct Products

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EN
The proper connection number of a graph is the least integer k for which the graph has an edge coloring with k colors, with the property that any two vertices are joined by a properly colored path. We prove that given two connected non-bipartite graphs, one of which is (vertex) 2-connected, the proper connection number of their direct product is 2.
EN
Let G = (V,E) be a graph. A total restrained dominating set is a set S ⊆ V where every vertex in V∖S is adjacent to a vertex in S as well as to another vertex in V∖S, and every vertex in S is adjacent to another vertex in S. The total restrained domination number of G, denoted by $γ_r^t(G)$, is the smallest cardinality of a total restrained dominating set of G. We determine lower and upper bounds on the total restrained domination number of the direct product of two graphs. Also, we show that these bounds are sharp by presenting some infinite families of graphs that attain these bounds.
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