Total domination versus paired-domination in regular graphs

A subset S of vertices of a graph G is a dominating set of G if every vertex not in S has a neighbor in S, while S is a total dominating set of G if every vertex has a neighbor in S. If S is a dominating set with the additional property that the subgraph induced by S contains a perfect matching, then S is a paired-dominating set. The domination number, denoted ?(G), is the minimum cardinality of a dominating set of G, while the minimum cardinalities of a total dominating set and paired-dominating set are the total domination number, ?_t(G), and the paired-domination number, ?_pr(G), respectively. For k = 2, let G be a connected k-regular graph. It is known [Schaudt, Total domination versus paired domination, Discuss. Math. Graph Theory 32 (2012) 435–447] that ?_pr(G)/?_t(G) = (2k)/(k + 1). In the special case when k = 2, we observe that ?_pr(G)/?_t(G) = 4/3, with equality if and only if G^~= C₅. When k = 3, we show that ?_pr(G)/?_t(G) = 3/2, with equality if and only if G is the Petersen graph. More generally for k = 2, if G has girth at least 5 and satisfies ?_pr(G)/?_t(G) = (2k)/(k + 1), then we show that G is a diameter-2 Moore graph. As a consequence of this result, we prove that for k = 2 and k = 57, if G has girth at least 5, then ?_pr(G)/?_t(G) = (2k)/(k + 1), with equality if and only if k = 2 and G ^~= C₅ or k = 3 and G is the Petersen graph.