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The design spectrum of a simple graph $G$ is the set of positive integers $n$ such that there exists an edgewise decomposition of the complete graph $K_n$ into $n(n - 1)/(2 |E(G)|)$ copies of $G$. We compute the design spectra for 7788 6-regular graphs with 12 vertices., Comment: 261 pages. The paper without the appendix will be submitted to a journal

We give direct constructions for 233 group divisible designs with block size five, mostly of type $g^u m^1$, $m > 0$., Comment: 63 pages. This is intended as a supplement to 'Group divisible designs with block size five', Australas. J. Combin. 87 (2023), 1-11

We report some group divisible designs with block size five, including types $6^{15}$ and $10^{15}$. As a consequence we are able to extend the known spectrum for 5-GDDs of type $g^u$., Comment: 11 pages. New result for this version: 5-GDD 3^45. The paper is under review with a journal

A generalized pentagonal geometry PENT($k$,$r$,$w$) is a partial linear space, where every line is incident with $k$ points, every point is incident with $r$ lines, and for each point, $x$, the set of points not collinear with $x$ forms the point set of a Steiner system $S(2,k,w)$ whose blocks are lines of the geometry. If $w = k$, the structure is called a pentagonal geometry and denoted by PENT($k$,$r$). The deficiency graph of a PENT($k$,$r$,$w$) has as its vertices the points of the geometry, and there is an edge between $x$ and $y$ precisely when $x$ and $y$ are not collinear. Our primary objective is to investigate generalized pentagonal geometries PENT($k$,$r$,$w$) where the deficiency graph has girth 4. We describe some construction methods, including a procedure that preserves deficiency graph connectedness, and we prove a number of theorems regarding the existence spectra for $k = 3$ and various values of $w$. In addition, we present some new PENT(4,$r$) (including PENT(4,25)) and PENT(5,$r$) with connected deficiency graphs. Consequently, we prove that there exist pentagonal geometries PENT($k$,$r$) with deficiency graphs of girth at least 5 for $r \ge 13$, $r$ congruent to 1 modulo 4 when $k = 4$, and for $r \ge 200000$, $r$ congruent to 0 or 1 modulo 5 when $k = 5$. We conclude with a discussion of appropriately defined identifying codes for pentagonal geometries., Comment: 84 pages. Abstract changed. Lemmas and theorems in Sections 4 and 5 significantly improved. Many more directly constructed PENT(4,r). New section: Identifying codes for pentagonal geometries. The paper with the appendix omitted will be submitted to a journal

A pentagonal geometry PENT($k$, $r$) is a partial linear space, where every line is incident with $k$ points, every point is incident with $r$ lines, and for each point $x$, there is a line incident with precisely those points that are not collinear with $x$. Here we generalize the concept by allowing the points not collinear with $x$ to form the point set of a Steiner system $S(2,k,w)$ whose blocks are lines of the geometry., Comment: 143 pages, 1 figure. An abridged version, with the Appendix omitted, will be submitted to a journal

New results on pentagonal geometries PENT(k,r) with block sizes k = 3 or k = 4 are given. In particular we completely determine the existence spectra for PENT(3,r) systems with the maximum number of opposite line pairs as well as those without any opposite line pairs. A wide-ranging result about PENT(3,r) with any number of opposite line pairs is proved. We also determine the existence spectrum of PENT(4,r) systems with eleven possible exceptions.

A pentagonal geometry PENT($k$, $r$) is a partial linear space, where every line, or block, is incident with $k$ points, every point is incident with $r$ lines, and for each point $x$, there is a line incident with precisely those points that are not collinear with $x$. An opposite line pair in a pentagonal geometry consists of two parallel lines such that each point on one of the lines is not collinear with precisely those points on the other line. We give a direct construction for an infinite sequence of pentagonal geometries with block size 3 and connected deficiency graphs. Also we present 39 new pentagonal geometries with block size 4 and five with block size 5, all with connected deficiency graphs. Consequentially we determine the existence spectrum up to a few possible exceptions for PENT(4, $r$) that do not contain opposite line pairs and for PENT(4, $r$) with one opposite line pair. More generally, given $j$ we show that there exists a PENT(4, $r$) with $j$ opposite line pairs for all sufficiently large admissible $r$. Using some new group divisible designs with block size 5 (including types $2^{35}$, $2^{71}$ and $10^{23}$) we significantly extend the known existence spectrum for PENT(5, $r$)., Comment: 42 pages. Abstract changed. Lemma 4.1, Theorem 4.2 and Table 8 updated. The paper will be submitted to a journal

The design spectrum has been determined for ten of the 15 graphs with six vertices and ten edges. In this paper we solve the design spectrum problem for the remaining five graphs with three possible exceptions., Comment: 26 pages, including 15-page Appendix. The paper will be submitted to a journal

We discuss group divisible designs with block size four and type $g^u b^1 (gu/2)^1$, where $u = 5$, 6 and 7. For integers $a$ and $b$, we prove the following. (i) A 4-GDD of type $(4a)^5 b^1 (10a)^1$ exists if and only if $a \ge 1$, $b \equiv a$ (mod 3) and $4a \le b \le 10a$. (ii) A 4-GDD of type $(6a+3)^6 b^1 (18a+9)^1$ exists if and only if $a \ge 0$, $b \equiv 3$ (mod 6) and $6a+3 \le b \le 18a + 9$. (iii) A 4-GDD of type $(6a)^6 b^1 (18a)^1$ exists if and only if $a \ge 1$, $b \equiv 0$ (mod 3) and $6a \le b \le 18a$. (iv) A 4-GDD of type $(12a)^7 b^1 (42a)^1$ exists if and only if $a \ge 1$, $b \equiv 0$ (mod 3) and $12a \le b \le 42a$, except possibly for $12a \in \{120, 180, 240, 360, 420, 720, 840\}$, $24a < b < 42a$, for $12a \in \{144, 1008\}$, $30a < b < 42a$, and for $12a \in \{168, 252, 336, 504, 1512\}$, $36a < b < 42a$., Comment: 290 pages, including 275-page appendix. To make the paper self-contained, some basic definitions, theorems and remarks have been copied from arXiv:1903.07064. The main results are of course distinct from those of arXiv:1903.07064. An abridged version of the paper (with the appendix omitted) will be submitted to a journal

We deal with group divisible designs that have block size 4 and group type g^u m^1, where g = 2 or 4 (mod 6). We show that the necessary conditions for the existence of a 4-GDD of type g^u m^1 are sufficient when g = 14, 20, 22, 26, 28, 32, 34, 38, 40, 44, 46, 50, 52, 58, 62, 68, 76, 88, 92, 100, 104, 116, 124, 136, 152, 160, 176, 184, 200, 208, 224, 232, 248, 272, 304, 320, 368, 400, 448, 464 and 496. Using these results we go on to show that the necessary conditions are sufficient for g = 2^t q^s, q = 19, 23, 25, 29, 31, s, t = 1, 2, ..., as well as for g = 2^t q, q = 2, 5, 7, 11, 13, 17, t = 1, 2, ..., with possible exceptions 56^9 m^1, 80^9 m^1 and 112^9 m^1 for a few large values of m., Comment: 128 pages, including 107-page Appendix. This is the third of three papers dealing with 4-GDDs of type g^u m^1. To make the paper self-contained we have included material from arXiv:1806.07491. However, the main results are distinct from those of arXiv:1806.07491. An abridged version will be submitted to a journal

We show that the necessary conditions for the existence of 4-GDDs of type g^u m^1 are sufficient for g congruent to 0 (mod h), h = 39, 51, 57, 69, 87, 93, and for g = 13, 17, 19, 23, 25, 29, 31 and 35. More generally, we show that for all g congruent to 3 (mod 6), the possible exceptions occur only when u = 8 and g is not divisible by any of 9, 15, 21, 33, 39, 51, 57, 69, 87 or 93. Consequently we are able to extend the known spectrum for g congruent to 1 and g congruent to 5 (mod 6). Also we complete the spectrum for 4-GDDs of type (3a)^4 (6a)^1 (3b)^1., Comment: 106 pages, including 91-page Appendix. Abridged submitted for publication

We solve the design spectrum problem for all theta graphs with 10, 11, 12, 13, 14 and 15 edges, Comment: 88 pages (including 74-page Appendix). Abridged version in Journal of Algorithms and Computation, volume 49

The design spectrum problem is solved for the thirty-eight 24-vertex non-trivial snarks., Comment: 74 pages. Abridged version (omitting the Appendix) accepted by Utilitas Mathematica

The main aim of this paper is to solve the design spectrum problem for Tietze's graph, the two 18-vertex Blanusa snarks, the six snarks on 20 vertices (including the flower snark J5), the twenty snarks on 22 vertices (including the two Loupekine snarks) and Goldberg's snark 3. Together with the Petersen graph (for which the spectrum has already been computed) this list includes all non-trivial snarks of up to 22 vertices. We also give partial results for a selection of larger graphs: the two Celmins-Swart snarks, the 26- and 34-vertex Blanusa snarks, the flower snark J7, the double star snark, Zamfirescu's graph, Goldberg's snark 5, the Szekeres snark and the Watkins snark., Comment: 92 pages. References updated, pictures added, typos corrected. Abridged version omitting all design details accepted by Utilitas Mathematica