A k-star is the complete bipartite graph 
. Let G and H be graphs, and let 
be a partial H-decomposition of G. A partial H-decomposition, 
, of another graph 
is called an embedding of 
provided that 
and G is a subgraph of 
. We find an embedding of a partial k-star decomposition of 
into a k-star decomposition of 
, where s is at most 
if k is odd, and 
if k is even.
The notion of a symmetric Hamiltonian cycle system (HCS) of a graph Γ has been introduced and studied by J. Akiyama, M. Kobayashi, and G. Nakamura [J Combin Des 12 (2004), 39–45] for 
, by R. A. Brualdi and M. W. Schroeder [J Combin Des 19 (2011), 1–15] for 
, and then naturally extended by V. Chitra and A. Muthusamy [Discussiones Mathematicae Graph Theory, to appear] to the multigraphs 
and 
. In each case, there must be an involutory permutation ψ of the vertices fixing all the cycles of the HCS and at most one vertex. Furthermore, for 
, this ψ should be precisely the permutation switching all pairs of endpoints of the edges of I.
An HCS is cyclic if it is invariant under some cyclic permutation of all the vertices. The existence question for a cyclic HCS of 
has been completely solved by Jordon and Morris [Discrete Math (2008), 2440–2449]—and we note that their cyclic construction is also symmetric for 
(mod 8). It is then natural to study the existence problem of an HCS of a graph or multigraph Γ as above which is both cyclic and symmetric. In this paper, we completely solve this problem: in the case of even order, the final answer is that cyclicity and symmetry can always cohabit when a cyclic solution exists. On the other hand, imposing that a cyclic HCS of odd order is also symmetric is very restrictive; we prove in fact that an HCS of 
with both properties exists if and only if 
is a prime.
Symmetric orthogonal arrays and mixed orthogonal arrays are useful in the design of various experiments. They are also a fundamental tool in the construction of various combinatorial configurations. In this paper, we investigated the mixed orthogonal arrays with four and five factors of strength two, and proved that the necessary conditions of such mixed orthogonal arrays are also sufficient with several exceptions and one possible exception.
A Γ-design of the complete graph 
is a set 
of subgraphs isomorphic to Γ (blocks) whose edge-sets partition the edge-set of 
. 
is balanced if the number of blocks containing x is the same number of blocks containing y for any two vertices x and y. 
is orbit-balanced, or strongly balanced, if the number of blocks containing x as a vertex of a vertex-orbit A of Γ is the same number of blocks containing y as a vertex of A, for any two vertices x and y and for every vertex-orbit A of Γ. We say that 
is degree-balanced if the number of blocks containing x as a vertex of degree d in Γ is the same number of blocks containing y as a vertex of degree d in Γ, for any two vertices x and y and for every degree d in Γ. An orbit-balanced Γ-design is also degree-balanced; a degree-balanced Γ-design is also balanced. The converse is not always true. We study the spectrum for orbit-balanced, degree-balanced, and balanced Γ-designs of 
when Γ is a graph with five vertices, none of which is isolated. We also study the existence of balanced (respectively, degree-balanced) Γ-designs of 
which are not degree-balanced (respectively, not orbit-balanced).
We show that every 
-relative difference set D in 
relative to 
can be represented by a polynomial 
, where 
is a permutation for each nonzero a. We call such an f a planar function on 
. The projective plane Π obtained from D in the way of M. J. Ganley and E. Spence (J Combin Theory Ser A, 19(2) (1975), 134–153) is coordinatized, and we obtain necessary and sufficient conditions of Π to be a presemifield plane. We also prove that a function f on 
with exactly two elements in its image set and 
is planar, if and only if, 
for any 
.
A twofold blocking set (double blocking set) in a finite projective plane Π is a set of points, intersecting every line in at least two points. The minimum number of points in a double blocking set of Π is denoted by τ2(Π). Let PG(2,q) be the Desarguesian projective plane over GF(q), the finite field of q elements. We show that if q is odd, not a prime, and r is the order of the largest proper subfield of GF(q), then τ2PG(2,q))≤ 2(q+(q-1)/(r-1)). For a finite projective plane Π, let 
denote the maximum number of classes in a partition of the point-set, such that each line has at least two points in some partition class. It can easily be seen that 
(⋆) for every plane Π on v points. Let 
, p prime. We prove that for 
, equality holds in (⋆) if q and p are large enough.
A k-cycle system of a multigraph G is an ordered pair (V, C) where V is the vertex set of G and C is a set of k-cycles, the edges of which partition the edges of G. A k-cycle system of 
is known as a λ-fold k-cycle system of order V. A k-cycle system of 
(V, C) is said to be enclosed in a k-cycle system of 
if 
and 
. We settle the difficult enclosing problem for λ-fold 5-cycle systems with u = 1.
Squashed 6-cycle systems are introduced as a natural counterpart to 2-perfect 6-cycle systems. The spectrum for the latter has been determined previously in [5]. We determine completely the spectrum for squashed 6-cycle systems, and also for squashed 6-cycle packings.
A symmetric nested orthogonal array, denoted by NOA
, is an OA
which contains an OA
as a subarray, where 
. Nested orthogonal arrays are useful in designing an experimental setup consisting of two experiments, the expensive one of higher accuracy being nested in a relatively less expensive one of lower accuracy. In this paper, some combinatorial constructions of nested orthogonal arrays are provided. By employing these constructions, the existence spectrum of NOA
is completely determined.
Blokhuis and Mazzocca (A. Blokhuis and F. Mazzocca, The finite field Kakeya problem (English summary). Building bridges. Bolyai Soc Math Stud 19 (2008) 205–218) provide a strong answer to the finite field analog of the classical Kakeya problem, which asks for the minimum size of a point set in an affine plane π that contains a line in every direction. In this article, we consider the related problem of minimal Kakeya sets, namely Kakeya sets containing no smaller Kakeya sets, and provide an interesting infinite family of minimal Kakeya sets that are not of extremal size.
In recent years, several methods have been proposed for constructing 
-optimal and minimax-optimal supersaturated designs (SSDs). However, until now the enumeration problem of such designs has not been yet considered. In this paper, 
-optimal and minimax-optimal k-circulant SSDs with 
6, 10, 14, 18, 22, and 26 runs, 
factors and 
are enumerated in a computer search. We have also enumerated all 
-optimal and minimax-optimal k-circulant SSDs with 
(mod 4) and 
. The computer search utilizes the fact that theses designs are equivalent to certain 1-rotational resolvable balanced incomplete block designs. Combinatorial properties of these resolvable designs are used to restrict the search space.
Necessary conditions for existence of a resolvable group divisible design (GDD) with block size 3 and type 
(a nearly Kirkman triple system, NKTS(v)), are 
and 
(mod 6). In this paper, we look at doubly resolvable NKTS(v)s; here we find that these necessary conditions are sufficient, except possibly for 64 values of v.
In this paper, generalizing the result in [9], I construct strongly regular Cayley graphs by using union of cyclotomic classes of 
and Gauss sums of index w, where 
is even. In particular, we obtain three infinite families of strongly regular graphs with new parameters.
Large sets of orthogonal arrays (LOA) have been used to construct resilient functions and zigzag functions by D. R. Stinson. In this paper, a special kind of LOA, strong double large sets of orthogonal arrays (SDLOA), is introduced and some constructions are provided. Meanwhile, a construction of multimagic squares based on SDLOAs is also given. As its application, it is proved that a t-multimagic square of order 
exists whenever q is a prime power and 
, 
, which improves a similar result by H. Derksen et al. from primes to prime powers in Amer. Math. Monthly (2007).
Nonuniform group divisible designs (GDDs) have been studied by numerous researchers for the past two decades due to their essential role in the constructions for other types of designs. In this paper, we investigate the existence problem of 
-GDDs of type 
for 
. First, we determine completely the spectrum of 
-GDDs of types 
and 
. Furthermore, for general cases, we show that for each 
and 
, a 
-GDD of type 
exists if and only if 
, 
and 
, except possibly for 
, 
and 
.
A decomposition of a complete graph 
into disjoint copies of a complete bipartite graph 
is called a 
-design of order n. The existence problem of 
-designs has been completely solved for the graphs 
for 
, 
for 
, K2, 3 and K3, 3. In this paper, I prove that for all 
, if there exists a 
-design of order N, then there exists a 
-design of order n for all 
(mod 
) and 
. Giving necessary direct constructions, I provide an almost complete solution for the existence problem for complete bipartite graphs with fewer than 18 edges, leaving five orders in total unsolved.
We investigate the cop number of graphs based on combinatorial designs. Incidence graphs, point graphs, and block intersection graphs are studied, with an emphasis on finding families of graphs with large cop number. We generalize known results on Meyniel extremal families by supplying bounds on the incidence graphs of transversal designs, certain G-designs, and BIBDs with 
Families of graphs with diameter 2, C4-free, and with unbounded chromatic number are described with the conjectured asymptotically maximum cop number.
Among all 2-
-designs, we characterize the Hermitian unitals by the existence of sufficiently many translations. In arbitrary 2-
-designs, each group of translations with given center acts semiregularly on the set of points different from the center.
A k-GDCD, group divisible covering design, of type 
is a triple 
, where V is a set of gu elements, 
is a partition of V into u sets of size g, called groups, and 
is a collection of k-subsets of V, called blocks, such that every pair of elements in V is either contained in a unique group or there is at least one block containing it, but not both. This family of combinatorial objects is equivalent to a special case of the graph covering problem and a generalization of covering arrays, which we call CARLs. In this paper, we show that there exists an integer 
such that for any positive integers g and 
, there exists a 4-GDCD of type 
which in the worst case exceeds the Schönheim lower bound by δ blocks, except maybe when (1) 
and 
, or (2) 
, 
, and 
or 
. To show this, we develop constructions of 4-GDCDs, which depend on two types of ingredients: essential, which are used multiple times, and auxiliary, which are used only once in the construction. If the essential ingredients meet the lower bound, the products of the construction differ from the lower bound by as many blocks as the optimal size of the auxiliary ingredient differs from the lower bound.
A Spernerk-partition system on a set X is a set of partitions of X into k classes such that the classes of the partitions form a Sperner set system (so no class from a partition is a subset of a class from another partition). These systems were defined by Meagher, Moura, and Stevens in [6] who showed that if 
, then the largest Sperner k-partition system has size 
. In this paper, we find bounds on the size of the largest Sperner k-partition system where k does not divide the size of X, specifically, we give upper and lower bounds when 
, 
and 
.
The problem of the existence of a decomposition of the complete graph 
into disjoint copies of 
has been solved for all admissible orders n, except for 27, 36, 54, 64, 72, 81, 90, 135, 144, 162, 216, and 234. In this paper, I eliminate 4 of these 12 unresolved orders. Let Γ be a 
-design. I show that 
divides 2k3 for some 
and that 
. I construct 
-designs by prescribing 
as an automorphism group, and show that up to isomorphism there are exactly 24 
-designs with 
as an automorphism group. Moreover, I show that the full automorphism group of each of these designs is indeed 
. Finally, the existence of 
-designs of orders 135, 162, and 216 follows immediately by the recursive constructions given by G. Ge and A. C. H. Ling, SIAM J Discrete Math 21(4) (2007), 851–864.
A cycle C in a graph G is extendable if there is some other cycle in G that contains each vertex of C plus one additional vertex. A graph is cycle extendable if every non-Hamilton cycle in the graph is extendable. A balanced incomplete block design, BIBD
, consists of a set V of v elements and a block set 
of k-subsets of V such that each 2-subset of V occurs in exactly λ of the blocks of 
. The block-intersection graph of a design 
is the graph 
having 
as its vertex set and such that two vertices of 
are adjacent if and only if their corresponding blocks have nonempty intersection. In this paper, we prove that the block-intersection graph of any BIBD
is cycle extendable. Furthermore, we present a polynomial time algorithm for constructing cycles of all possible lengths in a block-intersection graph.