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The generalized Oberwolfach problem asks for a factorization of the complete graph $K_v$ into prescribed $2$-factors and at most a $1$-factor. When all $2$-factors are pairwise isomorphic and $v$ is odd, we have the classic Oberwolfach problem, which was originally stated as a seating problem: given $v$ attendees at a conference with $t$ circular tables such that the $i$th table seats $a_i$ people and ${\sum_{i=1}^t a_i = v}$, find a seating arrangement over the $\frac{v-1}{2}$ days of the conference, so that every person sits next to each other person exactly once. In this paper we introduce the related {\em minisymposium problem}, which requires a solution to the generalized Oberwolfach problem on $v$ vertices that contains a subsystem on $m$ vertices. That is, the decomposition restricted to the required $m$ vertices is a solution to the generalized Oberwolfach problem on $m$ vertices. In the seating context above, the larger conference contains a minisymposium of $m$ participants, and we also require that pairs of these $m$ participants be seated next to each other for $\left\lfloor\frac{m-1}{2}\right\rfloor$ of the days. When the cycles are as long as possible, i.e.\ $v$, $m$ and $v-m$, a flexible method of Hilton and Johnson provides a solution. We use this result to provide further solutions when $v \equiv m \equiv 2 \pmod 4$ and all cycle lengths are even. In addition, we provide extensive results in the case where all cycle lengths are equal to $k$, solving all cases when $m\mid v$, except possibly when $k$ is odd and $v$ is even., Comment: 25 pages

Zero forcing is a combinatorial game played on a graph with the ultimate goal of changing the colour of all the vertices at minimal cost. Originally this game was conceived as a one player game, but later a two-player version was devised in-conjunction with studies on the inertia of a graph, and has become known as the $q$-analogue of zero forcing. In this paper, we study and compute the $q$-analogue zero forcing number for various families of graphs. We begin with by considering a concept of contraction associated with trees. We then significantly generalize an equation between this $q$-analogue of zero forcing and a corresponding nullity parameter for all threshold graphs. We close by studying the $q$-analogue of zero forcing for certain Kneser graphs, and a variety of cartesian products of structured graphs., Comment: 29 pages

A complementary Gray code for binary n-tuples is one that, when all the tuples are complemented, is identical to itself; this is equivalent to the complement of the first half of the code being identical to the second half. We generalize the notion of complementary to q-ary n-tuples, fixed size combinations of an n-set and permutations and, in each case, construct complementary Gray codes. We relax, as weakly as possible, the notions of complementary to cases where necessary conditions for existence are violated and construct Gray codes within the weakened definitions: these include binary n-tuples when n is odd and Lee metric q-ary n-tuples when n is odd and q is even. Finally a lemma used in the construction for permutations offers the first known cyclic Gray code for the permutations of a particular family of multisets., Comment: 22 pages, 14 Tables

A pair of planes, both projective or both affine, of the same order and on the same pointset are orthogoval if each line of one plane intersects each line of the other plane in at most two points. In this paper we prove new constructions for sets of mutually orthogoval planes, both projective and affine, and review known results that are equivalent to sets of more than two mutually orthogoval planes. We also discuss the connection between sets of mutually orthogoval planes and covering arrays., Comment: 20 pages

A \textbf{double-change covering design} (DCCD) is a $v$-set $V$ and an ordered list $\mathcal{L}$ of $b$ blocks of size $k$ where every pair from $V$ must occur in at least one block and each pair of consecutive blocks differs by exactly two elements. It is \textbf{minimal} if it has the fewest block possible and \textbf{circular} when the first and last blocks also differ by two elements. We give a recursive construction that uses 1-factorizations and expansion sets to construct a DCCD($v+\frac{v+k-2}{k-2},k,b+\frac{v}{k-2}\frac{v+k-2}{2k-4}$) from a DCCD($v,k,b$). We construct circular DCCD($2k-2,k,k-1$) and circular DCCD($2k-3,k,k-2$) from single change covering designs and determine minimal DCCD when $v=2k-2$. We use difference methods to construct five infinite families of minimal circular DCCD($c(4k-6)+1,k,c^2(4k-6)+c$) when $c\leq 5$ for any $k\geq 3$. The recursive construction is then used to build twelve additional minimal DCCD from members of these infinite families. Finally the difference method is used to construct a minimal circular DCCD(61,4,366)., Comment: 22 pages, 12 tables, 5 figures

A \textbf{single change covering design} is a $v$-set $X$ and an ordered list $\cL$ of $b$ blocks of size $k$ where every $t$-set must occur in at least one block. Each pair of consecutive blocks differs by exactly one element. A single change covering design is circular when the first and last blocks also differ by one element. A single change covering design is minimum if no other smaller design can be constructed for a given $v, k$. In this paper we use a new recursive construction to solve the existence of circular \sccd($v,4,b$) for all $v$ and three residue classes of circular \sccd($v,5,b$) modulo 16. We solve the existence of three residue classes of \sccd$(v,5,b)$ modulo 16. We prove the existence of circular \sccd$(2c(k-1)+1,k,c^2(2k-2)+c)$, for all $c\geq 1, k\geq2 $, using difference methods., Comment: 23 pages, 15 Tables

This work shows several direct and recursive constructions of ordered covering arrays using projection, fusion, column augmentation, derivation, concatenation and cartesian product. Upper bounds on covering codes in NRT spaces are also obtained by improving a general upper bound. We explore the connection between ordered covering arrays and covering codes in NRT spaces, which generalize similar results for the Hamming metric. Combining the new upper bounds for covering codes in NRT spaces and ordered covering arrays, we improve upper bounds on covering codes in NRT spaces for larger alphabets. We give tables comparing the new upper bounds for covering codes to existing ones., Comment: 27 pages

In this work, constructions of ordered covering arrays are discussed and applied to obtain new upper bounds on covering codes in Rosenbloom-Tsfasman spaces (RT spaces), improving or extending some previous results., Comment: 12 pages

We present a new construction of ordered orthogonal arrays (OOA) of strength $t$ with $(q + 1)t$ columns over a finite field $\mathbb{F}_{q}$ using linear feedback shift register sequences (LFSRs). OOAs are naturally related to $(t, m, s)$-nets, linear codes, and MDS codes. Our construction selects suitable columns from the array formed by all subintervals of length $\frac{q^{t}-1}{q-1}$ of an LFSR sequence generated by a primitive polynomial of degree $t$ over $\mathbb{F}_{q}$. We prove properties about the relative positions of runs in an LFSR which guarantee that the constructed OOA has strength $t$. The set of parameters of our OOAs are the same as the ones given by Rosenbloom and Tsfasman (1997) and Skriganov (2002), but the constructed arrays are different. We experimentally verify that our OOAs are stronger than the Rosenbloom-Tsfasman-Skriganov OOAs in the sense that ours are "closer" to being a "full" orthogonal array. We also discuss how our OOA construction relates to previous techniques to build OOAs from a set of linearly independent vectors over $\mathbb{F}_{q}$, as well as to hypergraph homomorphisms., Comment: 12 pages

In this paper, we investigate the randomness properties of sequences in $\mathbb{Z}_v$ derived from permutations in $\mathbb{Z}_{p}^*$ using the remainder function modulo $v$, where $p$ is a prime integer. Motivated by earlier studies with a cryptographic focus we compare sequences constructed from the ElGamal function $x \to g^x$ for $x\in\mathbb{Z}_{>0}$ and $g$ a primitive element of $\mathbb{Z}_{p}^*$, to sequences constructed from random permutations of $\mathbb{Z}_{p}^*$. We prove that sequences obtained from ElGamal have maximal period and behave similarly to random permutations with respect to the balance and run properties of Golomb's postulates for pseudo-random sequences. Additionally we show that they behave similarly to random permutations for the tuple balance property. This requires some significant work determining properties of random balanced periodic sequences. In general, for these properties and excepting for very unlikely events, the ElGamal sequences behave the same as random balanced sequences., Comment: 26 pages, 6 figures

A $(k,\ell)$-partition is a set partition which has $\ell$ blocks each of size $k$. Two uniform set partitions $P$ and $Q$ are said to be partially $t$-intersecting if there exist blocks $P_{i}$ in $P$ and $Q_{j}$ in $Q$ such that $\left| P_{i} \cap Q_{j} \right|\geq t$. In this paper we prove a version of the Erd\H{o}s-Ko-Rado theorem for partially $2$-intersecting $(k,\ell)$-partitions. In particular, we show for $\ell$ sufficiently large, the set of all $(k,\ell)$-partitions in which a block contains a fixed pair is the largest set of 2-partially intersecting $(k,\ell)$-partitions. For for $k=3$, we show this result holds for all $\ell$., Comment: 19 pages

We introduce an impartial combinatorial game on Steiner triple systems called Nofil. Players move alternately, choosing points of the triple system. If a player is forced to fill a block on their turn, they lose. We explore the play of Nofil on all Steiner triple systems up to order 15 and a sampling for orders 19, 21, and 25. We determine the optimal strategies by computing the nim-values for each game and its subgames. The game Nofil can be thought of in terms of play on a corresponding hypergraph. As game play progresses, the hypergraph shrinks and will eventually be equivalent to playing the game Node Kayles on an isomorphic graph. Node Kayles is well studied and understood. Motivated by this, we study which Node Kayles positions can be reached, i.e. embedded into a Steiner triple system. We prove necessary conditions and sufficient conditions for the existence of such graph embeddings and conclude that the complexity of determining the outcome of the game Nofil on Steiner triple systems is PSPACE-complete., Comment: 34 pages, 6 figures, 6 tables

In this paper we provide results on using integer programming (IP) and constraint programming (CP) to search for sets of mutually orthogonal latin squares (MOLS). Both programming paradigms have previously successfully been used to search for MOLS, but solvers for IP and CP solvers have significantly improved in recent years and data on how modern IP and CP solvers perform on the MOLS problem is lacking. Using state-of-the-art solvers as black boxes we were able to quickly find pairs of MOLS (or prove their nonexistence) in all orders up to ten. Moreover, we improve the effectiveness of the solvers by formulating an extended symmetry breaking method as well as an improvement to the straightforward CP encoding. We also analyze the effectiveness of using CP and IP solvers to search for triples of MOLS, compare our timings to those which have been previously published, and estimate the running time of using this approach to resolve the longstanding open problem of determining the existence of a triple of MOLS of order ten.

In 1989, computer searches by Lam, Thiel, and Swiercz experimentally resolved Lam's problem from projective geometry$\unicode{x2014}$the long-standing problem of determining if a projective plane of order ten exists. Both the original search and an independent verification in 2011 discovered no such projective plane. However, these searches were each performed using highly specialized custom-written code and did not produce nonexistence certificates. In this paper, we resolve Lam's problem by translating the problem into Boolean logic and use satisfiability (SAT) solvers to produce nonexistence certificates that can be verified by a third party. Our work uncovered consistency issues in both previous searches$\unicode{x2014}$highlighting the difficulty of relying on special-purpose search code for nonexistence results., Comment: To appear at the Thirty-Fifth AAAI Conference on Artificial Intelligence

In 1983, a computer search was performed for ovals in a projective plane of order ten. The search was exhaustive and negative, implying that such ovals do not exist. However, no nonexistence certificates were produced by this search, and to the best of our knowledge the search has never been independently verified. In this paper, we rerun the search for ovals in a projective plane of order ten and produce a collection of nonexistence certificates that, when taken together, imply that such ovals do not exist. Our search program uses the cube-and-conquer paradigm from the field of satisfiability (SAT) checking, coupled with a programmatic SAT solver and the nauty symbolic computation library for removing symmetries from the search., Comment: Appears in the Proceedings of the 31st International Workshop on Combinatorial Algorithms (IWOCA 2020)

In the 1970s and 1980s, searches performed by L. Carter, C. Lam, L. Thiel, and S. Swiercz showed that projective planes of order ten with weight 16 codewords do not exist. These searches required highly specialized and optimized computer programs and required about 2,000 hours of computing time on mainframe and supermini computers. In 2011, these searches were verified by D. Roy using an optimized C program and 16,000 hours on a cluster of desktop machines. We performed a verification of these searches by reducing the problem to the Boolean satisfiability problem (SAT). Our verification uses the cube-and-conquer SAT solving paradigm, symmetry breaking techniques using the computer algebra system Maple, and a result of Carter that there are ten nonisomorphic cases to check. Our searches completed in about 30 hours on a desktop machine and produced nonexistence proofs of about 1 terabyte in the DRAT (deletion resolution asymmetric tautology) format., Comment: To appear in Proceedings of the 29th International Joint Conference on Artificial Intelligence (IJCAI 2020)

Using techniques from the fields of symbolic computation and satisfiability checking we verify one of the cases used in the landmark result that projective planes of order ten do not exist. In particular, we show that there exist no projective planes of order ten that generate codewords of weight fifteen, a result first shown in 1973 via an exhaustive computer search. We provide a simple satisfiability (SAT) instance and a certificate of unsatisfiability that can be used to automatically verify this result for the first time. All previous demonstrations of this result have relied on search programs that are difficult or impossible to verify---in fact, our search found partial projective planes that were missed by previous searches due to previously undiscovered bugs. Furthermore, we show how the performance of the SAT solver can be dramatically increased by employing functionality from a computer algebra system (CAS). Our SAT+CAS search runs significantly faster than all other published searches verifying this result., Comment: To appear in Applicable Algebra in Engineering, Communication and Computing

Covering arrays are generalizations of orthogonal arrays that have been widely studied and are used in software testing. The probabilistic method has been employed to derive upper bounds on the sizes of minimum covering arrays and give asymptotic upper bounds that are logarithmic on the number of columns of the array. This corresponds to test suites with a desired level of coverage of the parameter space where we guarantee the number of test cases is logarithmic on the number of parameters of the system. In this paper, we study variable strength covering arrays, a generalization of covering arrays that uses a hypergraph to specify the sets of columns where coverage is required; (standard) covering arrays is the special case where coverage is required for all sets of columns of a fixed size $t$, its strength. We use the probabilistic method to obtain upper bounds on the number of rows of a variable strength covering array, given in terms of parameters of the hypergraph. We then compare this upper bound with another one given by a density-based greedy algorithm on different types of hypergraph such as $t$-designs, cyclic consecutive hypergraphs, planar triangulation hypergraphs, and a more specific hypergraph given by a clique of higher strength on top of a "base strength". The conclusions are dependent on the class of hypergraph, and we discuss specific characteristics of the hypergraphs which are more amenable to using different versions of the Lov\'{a}sz local lemma., Comment: revisions, primarily to Section 4

A covering array $\rm{CA}(N;t,k,v)$ of strength $t$ is an $N \times k$ array of symbols from an alphabet of size $v$ such that in every $N \times t$ subarray, every $t$-tuple occurs in at least one row. A covering array is \emph{optimal} if it has the smallest possible $N$ for given $t$, $k$, and $v$, and \emph{uniform} if every symbol occurs $\lfloor N/v \rfloor$ or $\lceil N/v \rceil$ times in every column. Prior to this paper the only known optimal covering arrays for $t=2$ were orthogonal arrays, covering arrays with $v=2$ constructed from Sperner's Theorem and the Erd\H{o}s-Ko-Rado Theorem, and eleven other parameter sets with $v>2$ and $N > v^2$. In all these cases, there is a uniform covering array with the optimal size. It has been conjectured that there exists a uniform covering array of optimal size for all parameters. In this paper a new lower bound as well as structural constraints for small uniform strength-$2$ covering arrays are given. Moreover, covering arrays with small parameters are studied computationally. The size of an optimal strength-$2$ covering array with $v > 2$ and $N > v^2$ is now known for $21$ parameter sets. Our constructive results continue to support the conjecture.

It has been conjectured that whenever an optimal covering array exists there is also a uniform covering array with the same parameters and this is true for all known optimal covering arrays. When used as a test suite, the application context may have pairs of parameters that must be avoided and Covering arrays avoiding forbidden edges (CAFE) are a generalization accommodating this requirement. We prove that there is an arc-transitive, highly symmetric constraint graph where the unique optimal covering array avoiding forbidden edges is not uniform. This does not refute the conjecture but it does show that placing even highly symmetric constraints on covering arrays can force non-uniformity of optimal arrays.

In \cite{Castoldi}, $q^t \by (q+1)t$ ordered orthogonal arrays (OOAs) of strength $t$ over the alphabet $\FF_q$ were constructed using linear feedback shift register sequences (LFSRs) defined by {\em primitive} polynomials in $\FF_q[x]$. In this paper we extend this result to all polynomials in $\FF_q[x]$ which satisfy some fairly simple restrictions, restrictions that are automatically satisfied by primitive polynomials. While these restrictions sometimes reduce the number of columns produced from $(q+1)t$ to a smaller multiple of $t$, in many cases we still obtain the maximum number of columns in the constructed OOA when using non-primitive polynomials. For small values of $q$ and $t$, we generate OOAs in this manner for all permissible polynomials of degree $t$ in $\FF_q[x]$ and compare the results to the ones produced in \cite{Castoldi}, \cite{Rosenbloom} and \cite{Skriganov} showing how close the arrays are to being "full" orthogonal arrays. Unusually for finite fields, our arrays based on non-primitive irreducible and even reducible polynomials are closer to orthogonal arrays than those built from primitive polynomials.

A locally irregular graph is a graph whose adjacent vertices have distinct degrees, a regular graph is a graph where each vertex has the same degree and a locally regular graph is a graph where for every two adjacent vertices u, v, their degrees are equal. In this work, we study the set of all problems which are related to decomposition of graphs into regular, locally regular and/or locally irregular subgraphs and we present some polynomial time algorithms, NP-completeness results, lower bounds and upper bounds for them. Among our results, one of our lower bounds makes use of mutually orthogonal Latin squares which is relatively novel., Comment: 31 pages, 8 figures

In this paper we introduce a natural mathematical structure derived from Samuel Beckett's play "Quad". We call this structure a binary Beckett-Gray code. We enumerate all codes for $n \leq 6$ and give examples for $n=7,8$. Beckett-Gray codes can be realized as successive states of a queue data structure. We show that the binary reflected Gray code can be realized as successive states of two stack data structures., Comment: 6 pages, 3 tables. Revisions requested from Journal of Combinatorial Mathematics and Combinatorial Computing

The de Bruijn torus (or grid) problem looks to find an $n$-by-$m$ binary matrix in which every possible $j$-by-$k$ submatrix appears exactly once. The existence and construction of these binary matrices was determined in the 70's, with generalizations to $d$-ary matrices in the 80's and 90's. However, these constructions lacked efficient decoding methods, leading to new constructions in the early 2000's. The new constructions develop cross-shaped patterns (rather than rectangular), and rely on a concept known as a half de Bruijn sequence. In this paper, we further advance this construction beyond cross-shape patterns. Furthermore, we show results for universal cycle grids, based off of the one-dimensional universal cycles introduced by Chung, Diaconis, and Graham, in the 90's. These grids have many applications such as robotic vision, location detection, and projective touch-screen displays., Comment: 16 pages, 4 figures; to appear in Discrete Mathematics

A covering array $CA(N; t,k,v)$ is an $N \times k$ array $A$ whose each cell takes a value for a $v$-set $V$ called an alphabet. Moreover, the set $V^t$ is contained in the set of rows of every $N \times t$ subarray of $A$. The parameter $N$ is called the size of an array and $CAN(t,k,v)$ denotes the smallest $N$ for which a $CA(N; t,k,v)$ exists. It is well known that $CAN(t,k,v) = {\rm \Theta}(\log_2 k)$~\cite{godbole_bounds_1996}. In this paper we derive two upper bounds on $d(t,v)=\limsup_{k \rightarrow \infty} \frac{CAN(t,k,v)}{\log_2 k}$ using the algorithmic approach to the Lov\'{a}sz local lemma also known as entropy compression., Comment: Submitted in January 2015

Karonski, Luczak, and Thomason (2004) conjectured that, for any connected graph G on at least three vertices, there exists an edge weighting from {1,2,3} such that adjacent vertices receive different sums of incident edge weights. Bartnicki, Grytczuk, and Niwcyk (2009) made a stronger conjecture, that each edge's weight may be chosen from an arbitrary list of size 3 rather than {1,2,3}. We examine a variation of these conjectures, where each vertex is coloured with a sequence of edge weights. Such a colouring relies on an ordering of the graph's edges, and so two variations arise -- one where we may choose any ordering of the edges and one where the ordering is fixed. In the former case, we bound the list size required for any graph. In the latter, we obtain a bound on list sizes for graphs with sufficiently large minimum degree. We also extend our methods to a list variation of irregularity strength, where each vertex receives a distinct sequence of edge weights., Comment: Accepted to Discrete Mathematics and Theoretical Computer Science

We consider the class of graphs for which the edge connectivity is equal to the maximum number of edge-disjoint spanning trees, and the natural generalization to matroids, where the cogirth is equal to the number of disjoint bases. We provide descriptions of such graphs and matroids, showing that such a graph (or matroid) has a unique decomposition. In the case of graphs, our results are relevant for certain communication protocols., Comment: 15 pages, 5 figures. Supersedes arXiv:1012.2478

We propose a network protocol similar to the $k$-tree protocol of Itai and Rodeh [{\em Inform.\ and Comput.}\ {\bf 79} (1988), 43--59]. To do this, we define an {\em $t$-uncovering-by-bases} for a connected graph $G$ to be a collection $\mathcal{U}$ of spanning trees for $G$ such that any $t$-subset of edges of $G$ is disjoint from at least one tree in $\mathcal{U}$, where $t$ is some integer strictly less than the edge connectivity of $G$. We construct examples of these for some infinite families of graphs. Many of these infinite families utilise factorisations or decompositions of graphs. In every case the size of the uncovering-by-bases is no larger than the number of edges in the graph and we conjecture that this may be true in general., Comment: 12 pages, 5 figures

We provide a structural description of, and invariants for, maximum spanning tree-packable graphs, i.e. those graphs G for which the edge connectivity of G is equal to the maximum number of edge-disjoint spanning trees in G. These graphs are of interest for the k-tree protocol of Itai and Rodeh [Inform. and Comput. 79 (1988), 43-59]., Comment: This paper has been superseded by arXiv:1203.1014 "On packing spanning trees in graphs and bases in matroids" by Robert F. Bailey, Mike Newman and Brett Stevens