An Application of Network Flows to Rearrangement of Series

For each permutation f of the set of positive integers, alltriples s, t, u are determined such that t and u are the lowerand upper limits of the sequence of partial sums of the ‘f-rearrangement’a_f(n) of some real series a_n with sum s.     

Eric Charles Milner

Eric Charles Milner was born on 17 May 1928 and brought up inLondon. His father was an engineer, but times were hard andwork was often difficult to obtain. So his mother had to helpout by working as a seamstress, and Eric was often looked afterby his grandmother. At the age of 11, he won a scholarship tothe Haberdashers' Aske's Boys' School, but never attended itin its permanent London buildings because the outbreak of theSecond World War caused all London schools and their pupilsto be evacuated to safer parts of the country. As a result,Eric, an only child and knowing none of his new schoolfellows,was billeted at a home near Reading where he was extremely unhappy.In despair, he ran away and returned to London, where, afterunsuccessful attempts to find him another billet, he roamedthe streets and missed school. After some time, he was eventuallyfound another billet where he received kindness and was muchhappier. Despite these disruptions and the other inevitableshortcomings of a war-time education, Eric's intelligence morethan sufficed to surmount such hurdles, and in later life hecould speak and write better than most of us. From 1946 to 1951, Eric attended King's College, London. Hegraduated with First Class Honours in 1949, when he was awardedthe Drew Gold Medal as the most distinguished Mathematics studentin that year, and a Research Studentship. He then studied foran MSc degree, taking ‘Modern algebra’ and ‘Quantummechanics (Wave mechanics)’ as his selected subjects,his supervisors being Richard Rado (then a Reader at King'sCollege) and Professor Charles Coulson. He received the MScdegree, with distinction, in 1950. This was followed by a year'sresearch in quantum mechanics under the supervision of ProfessorCoulson.     

Reconstruction of locally finite connected graphs with at least three infinite wings

Let G be a locally finite connected graph that can be expressed as the union of a finite subgraph and p disjoint infinite subgraphs, where 3 ≦ p < ∞, but cannot be expressed as the union of a finite subgraph and p + 1 disjoint infinite subgraphs. Then G is reconstructible.     

Reconstructing the number of copies of a valency-labeled finite graph in an infinite graph

Suppose that G, H are infinite graphs and there is a bijection Ψ; V(G) Ψ V(H) such that G - ξ ? H - Ψ(ξ) for every ξ ～ V(G). Let J be a finite graph and /(π) be a cardinal number for each π ? V(J). Suppose also that either /(π) is infinite for every π ? V(J) or J has a connected subgraph C such that /(π) is finite for every π ? V(C) and every vertex in V(J)/V(C) is adjacent to a vertex of C. Let (J, I, G) be the set of those subgraphs of G that are isomorphic to J under isomorphisms that map each vertex π of J to a vertex whose valency in G is /(π). We prove that the sets (J, I, G), m(J, I, H) have the same cardinality and include equal numbers of induced subgraphs of G, H respectively.     

More proofs of menger's theorem

Four ways of proving Menger's Theorem by induction are described. Two of them involve showing that the theorem holds for a finite undirected graph G if it holds for the graphs obtained from G by deleting and contracting the same edge. The other two prove the directed version of Menger's Theorem to be true for a finite digraph D if it is true for a digraph obtained by deleting an edge from D.     

Marriage in denumerable societies

A society is an ordered triple (M, W, K) of sets such that M, W are disjoint and K ? M × W. An espousal of (M, W, K) is a subset of K of the form {(a, e(a)) : a ∈ M} where e(a₁) ≠ e(a₂) whenever a₁ ≠ a₂. If M is countable, we associate with (M, W, K) and each ordinal α a function m_α from the set of subsets of W into the union of the set of integers and {? ∞, ∞}. Three different definitions of m_α (all fairly elaborate) are presented and their equivalence under suitable conditions is proved. Assuming M to be countable, we prove that (i) (M, W, K) has an espousal if and only if $\text{m}{}_{\mathrm{\Omega }}\text{(X) ? 0}$ for every subset X of W, where Ω is the first uncountable ordinal, and (ii) if X ? W and α ? β and m_α(X) < ∞ and m_α(Z) ? 0 for every subset Z of X then m_α(Z) = m_β(Z) for every subset Z of X. The result (i) is a theorem of Damerell and Milner, but the proof here presented differs somewhat in formulation and structure from theirs.     

The square of a block is Hamiltonian connected

Let B be a block (finite connected graph without cut-vertices) with at least four vertices and ξ, η be distinct vertices of B. We construct a new block M = M(B, ξ, η) containing five copies of B, and use the existence of a Hamiltonian circuit in M² to establish the existence of a Hamiltonian path starting at ξ and ending at η in B². A variant of this trick shows that B² ? ξ has a Hamiltonian circuit.