C. St. J. A. Nash-Williams

Amalgamations of almost regular edge-colourings of simple graphs

Abstract A finite graph F is a detachment of a finite graph G if G can be obtained from F by partitioning V(F) into disjoint sets S1, …, Sn and identifying the vertices in Si to form a single vertex αi for i = 1, …, n. Thus E(F) = E(G) and an edge which joins an element of Si to an element of Sj in F will join αi to αj in G. If L is a subset of E(G) then G(L) denotes the subgraph of G such that V(G(L)) = V(G), E(G(L)) = L. We call a graph almost regular if there is an integer d such that every vertex has valency d or d + 1. Suppose that E(G) is partitioned into disjoint sets E1, …, Er. Hilton [3] found necessary and sufficient conditions for the existence of a detachment F of G such that F is a complete graph with 2r + 1 vertices and F(Ei) is a Hamilton circuit of F for i = 1, …, r. We give a new proof of Hiltons theorem, which also yields a generalisation. Specifically, for any q ∈ {0, 1, …, r}, we find necessary and sufficient conditions for G to have a detachment F without loops or multiple edges such that F(E1), …, F(Er) are almost regular and F(E1), …, F(Eq) are 2-edge-connected and each vertex ξ of G arises by identification from a prescribed number g(ξ) of vertices of F.

Another Criterion for 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 ∈ e M } where E(a 1 ) ≠ E(a 2 ) whenever a 1 ≠ a 2 . For every transfinite sequence f of distinct elements of W , we define (in a somewhat complicated manner) a number q(f). We prove that a necessary and, if M is countable, sufficient condition for (M, W, K) to have an espousal is that q(f)⩾ )0 for every countable transfinite sequence f of distinct elements of W.

Reconstruction of infinite graphs

The paper recalls several known results concerning reconstruction and edge-reconstruction of infinite graphs, and draws attention to some possibly interesting unsolved problems.

An Application of Network Flows to Rearrangement of Series

For each permutation f of the set of positive integers, all triples s , t , u are determined such that t and u are the lower and upper limits of the sequence of partial sums of the ‘ f -rearrangement’ [sum ] a f ( n ) of some real series [sum ] a n with sum s .

More proofs of menger's theorem

Four ways of proving Mengers 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 Mengers Theorem to be true for a finite digraph D if it is true for a digraph obtained by deleting an edge from D.

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.

Another proof of a theorem concerning detachments of graphs

Let b be a positive integer-valued function on the set of vertices of a finite graph G. We give a new proof of a theorem which characterizes the least possible number of components of a graph obtainable from G by splitting each vertex ξ into b(ξ) vertices.

Reconstruction of locally finite connected graphs with two infinite wings

Abstract Let G be a locally finite connected graph which can be expressed as the union of a finite subgraph and two disjoint infinite subgraphs but cannot be expressed as the union of a finite subgraph and three disjoint infinite subgraphs. Then G is reconstructible.

Infinite digraphs with nonreconstructible outvalency sequences

We show that the outvalency sequence of an infinite digraph is not, in general, 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.