There are ten bipartite cubic graphs of order n ≤ 12. For each such graph G we give necessary and sufficient conditions for the existence of decompositions of Kn and of Km,n into copies of G.

Let V be a finite set of positive integers with sum equal to a multiple of the integer b . When does V have a partition into b parts so that all parts have equal sums? We develop algorithmic constructions which yield positive, albeit incomplete, answers for the following classes of set V , where n is a given positive integer: (1) an initial interval { a ∈ ℤ + : a ≤ n } ; (2) an initial interval of primes { p ∈ ℙ : p ≤ n } , where ℙ is the set of primes; (3) a divisor set { d ∈ ℤ + : d | n } ; (4) an aliquot set { d ∈ ℤ + : d | n , d < n } . Open general questions and conjectures are included for each of these classes.

A subset of an abelian group is midpoint-free if it contains no three distinct elements a, b, c such that a + b = 2c. We study midpoint-free sets in various classical topological groups. For every infinite cardinal kappa <= c, we show that the real line can be partitioned into K-many maximal midpoint-free sets. Examples of closed maximal midpoint-free subsets are given for topological groups such as R, C, S-1, S-1 x S-1. Finally, among sets that are not regular, such as nonmeasurable sets, Bernstein sets, and Luzin sets, we study instances which are midpoint-free.

How should mathematical objects be counted? Several quite distinct methods are popular including summation of finite series, use .of one-to-one correspondences, manipulation of generating functions, and application of the inclusion-exclusion principle, to name only the most common. Perhaps summation of finite series is the method used most widely, even though it tends to be a brute force approach relying on well-developed techniques of manipulation rather than on features which are inherent in the context of the particular problem. Its use reflects the viewpoint that the answer is of primary importance, while the way it is obtained is incidental: in other words, the end justifies the means. We are sure many besides ourselves have had the experience of solving a counting problem by summation of finite series and upon finding that the answer was relatively simple, have thought: There must be a more natural way to derive this solution! Our intention here is to show that the use of correspondences may provide more natural and more enlightening solutions to counting problems, by making use of inherent structural features. As a paradigm, we take the problem of counting all nondecreasing sequences of k natural numbers, none of which exceeds n: that is, the problem of determining the cardinality of the set

A set S⊂Z is midpoint-free if no ordered triple a,b,c∈S3 satisfies a+c=2b and a

Let $G$ be the unit distance graph in the plane. A well-known problem in combinatorial geometry is that of determining the chromatic number of $G$. It is known that $4\le \chi(G)\le 7$. The upper bound of 7 is obtained using tilings of the plane. The present paper studies two problems where we seek proper colourings of $G$, adding restrictions inspired by tilings: Let $H(\epsilon)$ be the graph whose vertices are the points of ${\mathbb R}^2$, with an edge between two points if their distance lies in the interval $[1,1+\epsilon]$. We show that for small $\epsilon$, $0<\epsilon\le \frac{3\sqrt{2}}{4}-1$, we have $6\le \chi(H(\epsilon))\le 7$. This improves the result of Exoo and Grytczuk et al. that $5\le \chi(H(\epsilon))$ for small $\epsilon$. Suppose that $G$ is properly coloured, but so that two solidly coloured regions meet along a straight line in some neighbourhood. Then at least 5 colours must be used.

A set of reals S⊂R is midpoint-free if it has no subset a, b, c⊆S such that a

Let K (r) n be the order n uniform complete multigraph with edge multiplicity r. A spanning tree decomposition of K (r) n partitions its edge set into a family T of edge-induced spanning trees. In a purely heterogeneous decomposition T no trees are isomorphic. Every order n tree occurs in a fully heterogeneous decomposition T. All trees have equal multiplicity in a balanced decomposition T. We say that T is reducible if it has a proper subfamily T′ ⊂ T such that T′ is a spanning tree decomposition of K n for some s with r > s > 0. We show that for fixed n and sufficiently large r, every decomposition of K (r) n is reducible. We also show that when n ≥ 6, K n has a purely heterogeneous decomposition T comprising a path, three trees of maximum degree Δ = 3, and one for each Δ > 3. 1 Heterogeneous Decompositions into Spanning Trees This paper continues the work done in [1] to examine heterogeneous graph decomposition involving families of spanning trees. Typically the graphs we decompose have at least one edge between each pair of vertices, so they are “over-complete” multigraphs, the multiplicity of each adjacency being at least one. The focus is on decompositions of uniform complete multigraphs, that is, complete multigraphs in which all adjacencies have the same multiplicity. A spanning tree decomposition of a multigraph K is a family T of edge-induced spanning trees that partitions the edges of K. A given multigraph K need not have any such decomposition. The purely heterogeneous case is when T is a set, so no two members of T are isomorphic. Typically T is a multiset; it is a homogeneous decomposition of K if all members are isomorphic, or a heterogeneous decomposition if at least two members are non-isomorphic. We denote by T(n) the family of all unlabeled trees of order n. If K has order n, a spanning tree decomposition T of K is fully heterogeneous if T(n) ⊆ T. Two spanning tree decompositions T and T′ of K are similar if each tree in T(n) has the same multiplicity in T and in T′, and they are equivalent if T can be transformed into T′ by some automorphism of K. Explanation of several of our notational conventions is appropriate. If T is a set or multiset and k is a positive integer, then kT is the multiset in which the multiplicity of each member is k times its multiplicity in T. If G and H are two simple graphs or multigraphs, the union G ∪ H comprises one copy of each, and the copies are vertex-disjoint. In contrast, the sum G + H comprises one copy of each, and the copies are edge-disjoint but the vertices of H are identified with distinct vertices of G (assuming the order of G is not less than the order of H). Usually the sum notation is ambiguous without further specification, because the vertex identifications can be achieved in various ways, but there is no ambiguity if G is a complete graph or, more generally, a uniform complete multigraph. In particular, if G = H we write 2G for the union and G for the sum with corresponding vertices identified. More generally, if k is any positive integer, then kG is the union of k copies of G, and G is the sum of k copies of G with corresponding vertices identified. (We read G as PURELY HETEROGENEOUS DECOMPOSITIONS 147 “G, k-fold”.) Again, the difference G−H is defined when G and H are simple graphs or multigraphs, with H a subgraph or submultigraph of G: the vertices of G − H are those of G, and the multiplicity of any adjacency in G−H is its multiplicity in G reduced by the corresponding multiplicity in H. The complete multigraphK (r) n is the order n uniform complete multigraph with all adjacencies of multiplicity r. The following two fully heterogeneous decompositions involving particular complete multigraphs are presented in [2], together with some oriented analogs: Theorem 1.1. The complete multigraph K (2) 6 can be decomposed into T(6), one copy of each of the six trees of order 6. Theorem 1.2. The complete multigraph K (2) 4 can be decomposed into 2T(4), two copies of each of the two trees of order 4. In [1], we studied the fully heterogeneous problem for K 5 and K 4 5 . The results are summarized in the following: Theorem 1.3. [1] The complete multigraph K (2) 5 has exactly 24 inequivalent fully heterogeneous spanning tree decompositions. There are 11 of type [2, 2, 1], and 13 of type [3, 1, 1]. Those of type [2, 2, 1] comprise two in the similarity class (2, 1, 2), and nine in the similarity class (1, 2, 2). Those of type [3, 1, 1] comprise six in the similarity class (1, 3, 1), and seven in the similarity class (1, 1, 3). Theorem 1.4. [1] The complete multigraph K (4) 5 has 34 similarity classes of fully heterogeneous spanning tree decompositions; the only potential classes not realized are (7, 2, 1) and (8, 1, 1). With (4, 5, 1) as the sole exception, 33 of the similarity classes contain reducible decompositions. Of these, 30 contain heterogeneously reducible decompositions, and exactly nine of those contain bi-heterogeneously reducible decompositions. The three similarity classes which contain reducible decompositions, but none that is heterogeneously reducible, are (5, 2, 3), (5, 3, 2) and (5, 4, 1). These are the motivating paradigms for the present paper. Here we investigate the possibility of various decompositions like these two, for the most part involving uniform complete multigraphs of order 5, but subsequently looking at some order n general decomposition results. We are planning a sequel paper to discuss the rich decomposition results in the oriented case. 2 Reducible Spanning Tree Decompositions of K (r) n We now consider reducibility of spanning tree decompositions in a more general context. Let n and r be any positive integers such that there exists a spanning tree decomposition T ofK (r) n . We say that T is reducible if it has a proper subfamily T′ ⊂ T such that T′ is a spanning tree decomposition of K n for some s with r > s > 0; such a subfamily T′ is a reduction of T. If T′ is a reduction of T, and T(n) ⊆ T′, then T′ is a fully heterogeneous reduction of T(n). We shall prove that if n is fixed and 148 ABUEIDA, BLINCO, CLARK, DAVEN AND EGGLETON r is sufficiently large, every spanning tree decomposition T of K (r) n is reducible, and every fully heterogeneous T has a fully heterogeneous reduction. We begin with some notation and terminology required in the proof of the key lemma. For any integer n ≥ 1, we denote the interval {i ∈ Z : 1 ≤ i ≤ n} by [1, · · · , n]. Let (Z) be the dominance poset on n-tuples of non-negative integers, with the dominance partial ordering: if α = [a1, · · · , an], β = [b1, · · · , bn] ∈ (Z) then α > β provided ai ≥ bi for each i ∈ [1, · · · , n] and strict inequality holds for at least one i. As usual, a non-empty subset A ⊂ (Z) is an independent set (or antichain) if no member of A dominates any other member of A. Lemma 2.1. Every independent subset of (Z) is finite. Proof. Let A be any non-empty independent subset of (Z). Choose any δ = [d1, · · · , dn] ∈ (Z). For any non-empty subset S ⊆ [1, · · · , n], define the subset AS(δ) = {α = [a1, · · · , an] ∈ A : ai = di for each i ∈ [1, · · · , n]\S} ⊆ A. We call AS(δ) the k-parameter subset of A determined by δ and S, where k = |S|. We shall prove that the k-parameter subsets of A are finite, for each k ∈ [1, · · · , n]. But A is an n-parameter subset of itself, namely A[1,··· ,n](δ) = A for any δ ∈ (Z), so it will follow that A is finite. Fix α ∈ A. If S is a singleton, say S = {r}, then α is a member of the 1-parameter subset Ar(α), where we write r to denote the subscript {r}. If β = [b1, · · · , bn] is any other member of Ar(α), then ar = br so either ar > br or br > ar. Then α > β or β > α, contradicting the independence of A. Hence Ar(α) = {α}. Now fix some k ∈ [1, · · · , n] with k < n, and suppose every k-parameter subset of A is finite. Choose any S ⊆ [1, · · · , n] with |S| = k + 1. Clearly α ∈ AS(α). If AS(α)\{α} is non-empty, choose any other member β ∈ AS(α). Since β is independent of α, there is at least one r ∈ S such that br < ar. For any s ∈ Z let α(r, s) ∈ (Z) be the n-tuple derived from α by replacing its rth entry ar by s. Then β ∈ AS\{r}(α(r, br)). Clearly α ∈ AS\{r}(α(r, br)), since α(r, ar) = α, so it follows without exception that

We obtain two identities and an explicit formula for the number of homomorphisms of a finite path into a finite path. For the number of endomorphisms of a finite path these give over-count and under-count identities yielding the closed-form formulae of Myers. We also derive finite Laurent series as generating functions which count homomorphisms of a finite path into any path, finite or infinite.

How many times must a die be thrown to get four consecutive sixes? More generally, and more precisely, Q1. What is the expected number of times we must throw a die to get n consecutive sixes? The answer, rather suggestively for the numerologically inclined, turns out to be A1. The expected number of throws is 6 + 62 + 63 + … + 6n. In some well-known dice games players throw several dice at the same time: in craps a pair (brace) of dice is thrown, while in yahtzee, yam and balut, up to five dice are thrown at a time. However, Q1 is not the same as

SummaryMathematical elegance is illustrated by strikingly parallel versions of the product and quotient rules of basic calculus, with some applications. Corresponding rules for second derivatives are given: the product rule is familiar, but the quotient rule is less so.

Classifying the positive integers as primes, composites, and the unit, is so familiar that it seems inevitable. However, other classifications can bring interesting relationships to our attention. In that spirit, let us classify positive integers by the number o? principal divisors they possess, where we define a principal divisor of a positive integer n to be any prime-power divisor pa \ n which is maximal (so p is prime, a is a positive integer, and pa+l is not a divisor of n). The standard notation pa \\n can be read as "/?fl is a principal divisor of n." The Fundamental Theorem of Arithmetic is usually stated in a form emphasizing how primes enter the structure of the positive integers, such as: Every positive integer is the product of a unique finite multiset of primes. (Recall that a multiset is a collection of elements in which multiple occurrences are permitted.) Alternatively, the Fundamental Theorem of Arithmetic can be stated in a form that focuses on how maximal prime powers enter the structure of the positive integers, such as: Every positive integer is the product of a unique finite set of powers of distinct primes. Consequently every positive integer is the product of its principal divisors, and every finite set of powers of distinct primes is the set of principal divisors of a unique positive integer. Of course, the number of principal divisors of n is equal to the number of distinct prime factors of n, but here the principal divisors are the simple structural components of n, whereas the distinct prime factors are but a shadow of that structure. Readers who find the present paper of interest might find similar interest in [6], where upper bounds on the sum of principal divisors of n are established by elementary means. For each integer n > 0, let Pn be the set of all positive integers with exactly n principal divisors, so Pq = {1}, and

The level set $G(n,m)$ comprises all unlabelled simple graphs of order $n$ and size $m$, and is partitioned into similarity classes, comprising all graphs with the same degree sequence. When graphs are ordered lexicographically by their signature, a unique numerical list of structural descriptors, the similarity classes of $G(n,m)$ occur in contiguous blocks; the ¯rst graph in each similarity class is its sentinel. The sentinel of the ¯rst similarity class in each $G(n,m)$ is determined, and shown to be the unique realization of its degree sequence. The degree sequence of the last similarity class in each $G(n,m)$ is also determined, as are the exact size range for which it has more than one realization, and the exact size range for which its sentinel has more than one component.