Paratopism is a well‐known action of the wreath product scriptSn≀scriptS3 on Latin squares of order n. A paratopism that maps a Latin square to itself is an autoparatopism of that Latin square. Let Par(n) denote the set of paratopisms that are an autoparatopism of at least one Latin square of order n. We prove a number of general properties of autoparatopisms. Applying these results, we determine Par(n) for n⩽17. We also study the proportion of all paratopisms that are in Par(n) as n→∞.

Suppose that n ≡ ±1 mod 6 and n 7. We construct a Latin square L n of order n with the following properties:r L n has no proper subsquares of order 3 or more. r L n has exactly one intercalate (subsquare of order 2). r When the intercalate is replaced by the other possible subsquare on the same symbols, the resulting Latin square is in the same species as L n .Hence L n generalizes the square that Sade famously found to complete Norton's enumeration of Latin squares of order 7. In particular, L n is what is known as a self-switching Latin square and possesses a near-autoparatopism.

Abstract. Let j ≥ 3. Given that m j (H,G) denotes the smallest positive integer s such that K j×s →(H,G). In this paper, we exhaustively find m j (P 4 ,G) for all 11 non-isomorphic graphs G on 4 vertices, out of which 6 graphs G are connected and the others are disconnected.

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