A vertex x in a digraph D is said to resolve a pair u, v of vertices of D if the distance from u to x does not equal the distance from v to x. A set S of vertices of D is a resolving set for D if every pair of vertices of D is resolved by some vertex of S. The smallest cardinality of a resolving set for D, denoted by dim(D), is called the metric dimension for D. Sharp upper and lower bounds for the metric dimension of the Cayley digraphs Cay( : ), where is the group Z n 1 ⊕ Z n 2 ⊕ · · · ⊕ Z n m and is the canonical set of generators, are established. The exact value for the metric dimension of Cay({(0, 1), (1, 0)} : Z n ⊕ Z m ) is found. Moreover, the metric dimension of the Cayley digraph of the dihedral group D n of order 2n with a minimum set of generators is established. The metric dimension of a (di)graph is formulated as an integer programme. The corresponding linear programming formulation naturally gives rise to a fractional version of the metric dimension of a (di)graph. The fractional dual implies an integer dual for the metric dimension of a (di)graph which is referred to as the metric independence of the (di)graph. The metric independence of a (di)graph is the maximum number of pairs of vertices such that no two pairs are resolved by the same vertex. The metric independence of the n-cube and the Cayley digraph Cay( : D n ), where is a minimum set of generators for D n , are established.

Let G be a (di)graph and S a set of vertices of G. We say S resolves two vertices u and v of G if d(u, S) = d(v, S). A partition Π = {P 1 , P 2 , . . . , P k } of V (G) is a resolving partition of G if every two vertices of G are resolved by P i for some i (1 ≤ i ≤ k). The smallest cardinality of a resolving partition of G, denoted by pd(G), is called the partition dimension of G. A vertex r of G resolves a pair u, v of vertices of G if d(u, r) = d(v, r). A set R of vertices of G is a resolving set for G if every two vertices of G are resolved by some vertex of R. The smallest cardinality of a resolving set of vertices, denoted by dim(G), is called the metric dimension of G. We begin by disproving a conjecture made by Chartrand, Salehi and Zhang regarding the partition dimension of products of graphs. The partition dimension of Cayley digraphs of abelian groups with a specific minimal set of generators is shown to be at most one more than the number of generators with equality for one or two generators. It is known that pd(G) ≤ dim(G)+1. It is pointed out that for every positive integer M there are Cayley digraphs D for which dim(D) − pd(D) ≥ M , and that there are classes of Cayley digraphs D such that pd(D) dim(D) → 0 as |V (D)| → ∞. Moreover, it is shown that the partition dimension of the Cayley digraph for the dihedral group of order 2n (n ≥ 3) with a minimum set of generators is 3. We conclude by introducing a more general class of problems for which the problems of finding the metric dimension and partition dimension of a (di)graph are the two extremes and provide an interpretation of the transition between these two invariants. (2000). 05C12, 05C20, 05C90. Mathematics Subject Classification

Abstract. Let G = (V, E) be a connected graph (or hypergraph) and let d(x, y) denote the distance between verticesThe minimum cardinality of a resolving set for G is called the metric dimension of G, denoted by β(G). The circulant graph Cn(1, 2, . . . , t) has vertex set {v0, v1, . . . , vn−1} and edges vivi+j where 0 ≤ i ≤ n − 1 and 1 ≤ j ≤ t and the indices are taken modulo n (2 ≤ t ≤ n 2 ). In this paper we determine the exact metric dimension of the circulant graphs Cn(1, 2, . . . , t), extending previous results due to Borchert and Gosselin (2013), Grigorious et al. (2014), andVetrík (2016). In particular, we show that β (Cn(1, 2, . . . , t)) = β (Cn+2t(1, 2, . . . , t)) for large enough n, which implies that the metric dimension of these circulants is completely determined by the congruence class of n modulo 2t. We determine the exact value of β(Cn (1, 2, . . . , t)) for n ≡ 2 mod 2t and n ≡ (t + 1) mod 2t and we give better bounds on the metric dimension of these circulants for n ≡ 0 mod 2t and n ≡ 1 mod 2t. In addition, we bound the metric dimension of Cartesian products of circulant graphs.

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For a positive integer $q$, a $k$-uniform hypergraph $X=(V,E)$ is $q$-complementary if there exists a permutation $\theta$ on $V$ such that the sets $E, E^{\theta}, E^{\theta^2},\ldots, E^{\theta^{q-1}}$ partition the set of $k$-subsets of $V$. The permutation $\theta$ is called a $q$-antimorphism of $X$. The well studied self-complementary uniform hypergraphs are 2-complementary. For an integer $n$ and a prime $p$, let $n_{(p)}=\max\{i:p^i \text{divides} n\}$. In this paper, we prove that a vertex-transitive $q$-complementary $k$-hypergraph of order $n$ exists if and only if $n^{n_{(p)}}\equiv 1 (\bmod q^{\ell+1})$ for every prime number $p$, in the case where $q$ is prime, $k = bq^\ell$ or $k=bq^{\ell}+1$ for a positive integer $b < k$, and $n\equiv 1(\bmod q^{\ell+1})$. We also find necessary conditions on the order of these structures when they are $t$-fold-transitive and $n\equiv t (\bmod q^{\ell+1})$, for $1\leq t < k$, in which case they correspond to large sets of isomorphic $t$-designs. Finally, we use group theoretic results due to Burnside and Zassenhaus to determine the complete group of automorphisms and $q$-antimorphisms of these hypergraphs in the case where they have prime order, and then use this information to write an algorithm to generate all of these objects. This work extends previous, analagous results for vertex-transitive self-complementary uniform hypergraphs due to Muzychuk, Potočnik, Šajna, and the author. These results also extend the previous work of Li and Praeger on decomposing the orbitals of a transitive permutation group.

In 2007, Szyma«ski and Wojda proved that for positive integers n, k with k ≤ n, a self-complementary k-uniform hypergraph of order n exists if and only if n k is even. In this paper, we characterize the cycle type of a k-complementing permutation in Sym(n) which has order equal to a power of 2. This yields a test to determine if a nite permutation is a k-complementing permutation, and an algorithm for generating all selfcomplementary k-hypergraphs of order n, up to isomorphism, for feasible n. We also obtain an alternative description of the necessary and sucient conditions on the order of a self-complementary k-uniform hypergraph, in terms of the binary representation of k. This extends previous results for the cases k = 2, 3, 4 due to Ringel, Sachs, Suprunenko, Kocay and Szyma«ski.

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In this paper, we examine the possible orders of t-subset-regular selfcomplementary k-uniform hypergraphs, which form examples of large sets of two isomorphic t-designs. We reformulate Khosrovshahi and TayfehRezaie's necessary conditions on the order of these structures in terms of the binary representation of the rank k, and these conditions simplify to a more transparent relation between the order n and rank k in the case where k is a sum of consecutive powers of 2. Moreover, we present new constructions for 1-subset-regular self-complementary uniform hypergraphs, and prove that these necessary conditions are sucient for all k, in the case where t = 1.