Wagner's theorem for Torus graphs

Wagner's theorem (any two maximal plane graphs having p vertices are equivalent under diagonal transformations) is extended to maximal torus graphs, graphs embedded in the torus with a maximal set of edges present. Thus any maximal torus graph having p vertices may be diagonally transformed into any other maximal torus graph having p vertices. As with Wagner's theorem, a normal form representing an intermediate stage in the above transformation is displayed. This result, along with Wagner's theorem, may make possible constructive characterizations of planar and toroidal graphs, through a wholly combinatorial definition of diagonal transformation.     

Merten's theorem for arithmetic progressions

Let k be an integer ≥ 1 and let l be an integer such that 1 ≤ l ≤ k, (l,k) = 1. An asymptotic formula (valid for large x) is obtained for the product

$\text{∏}\text{p≤x,p≡l(}\text{mod}\text{k)}\text{1?}\text{1}\text{p}$

, generalizing a familiar result of Mertens.     

Lloyd's theorem for perfect codes

We give another proof of Lloyd's theorem using homogeneous distance enumerators, and show that the same techniques will give similar theorems for more general types of codes. The theorem has been proved earlier by Delsarte and Lenstra. We thought it interesting that the result can be derived from some elementary polynomial manipulations. The methods and results herein should be considered as belonging to combinatorial coding theory, since it is not necessary to use the finite field approach to get them.     

Bernstein's theorem for compact groups

In this paper we generalize the classical Bernstein theorem concerning the absolute convergence of the Fourier series of Lipschitz functions. More precisely, we consider a group G which is finite dimensional, compact, and separable and has an infinite, closed, totally disconnected, normal subgroup D, such that $\text{G}\text{D}$ is a Lie group. Using this structure, we define in a natural way the notion of Lipschitz condition, and then prove that a function which satisfies a Lipschitz condition of order greater than $\text{(}\text{dim}\text{G + 1)}\text{2}$ belongs to the Fourier algebra of G.     

Clark's theorem for semi-infinite convex programs

In 1961, Clark proved that if either the feasible region of a linear program or its dual is nonempty and bounded, then the other is unbounded. Recently, Duffin has extended this result to a convex program and its Lagrangian dual. Moreover, Duffin showed that under this boundedness assumption there is no duality gap. The purpose of this paper is to extend Duffin's results to semi-infinite programs.     

On Petersen's graph theorem

In this paper we prove the following: let G be a graph with e_G edges, which is (k ? 1)-edge- connected, and with all valences ?k. Let 1?r?k be an integer, then G contains a spanning subgraph H, so that all valences in H are ?r, with no more than ?re_G?k? edges. The proof is based on a useful extension of Tutte's factor theorem [4,5], due to Lovász [3]. For other extensions of Petersen's theorem, see [6,7,8].     

On Dilworth's decomposition theorem

For every countable po set P without infinite chains there exists a partition (C_j: j?I) of P into pairwise disjoint chains and an antichain A such that C_j∩A≠Ø for every i?I.     

On Schmerl's effective version of Brooks' theorem

A new proof is given of Schmerl's recent result that a highly recursive graph G with χ(G) ≤ k according to Brooks' theorem, has a recursive k-colouring.     

Green's theorem and Gorenstein sequences

We study consequences, for a standard graded algebra, of extremal behavior in Green's Hyperplane Restriction Theorem. First, we extend his Theorem 4 from the case of a plane curve to the case of a hypersurface in a linear space. Second, assuming a certain Lefschetz condition, we give a connection to extremal behavior in Macaulay's theorem. We apply these results to show that $(1,19,17,19,1)$ is not a Gorenstein sequence, and as a result we classify the sequences of the form $(1,a,a?2,a,1)$ that are Gorenstein sequences.     

A refinement of Kuratowski's theorem

The conjecture of Kelmans that any 3-connected non-planar graph with at least six vertices contains a cycle with three pairwise crossing chords is proved. Using this, a refinement of Kuratowski's theorem which also includes the result of Tutte that a graph is planar if and only if every cycle has a bipartite overlap graph is obtained.     

A note on Fung-Fu's theorem

The axiomatic approach to rational decision malting in a fuzzy environment proposed by Fung and Fu (1975) lays down a too restrictive family of aggregation rules. Because of such axioms, the independence of the size of the groups under aggregation is assumed; moveover, in a general problem, these groups are fuzzy sets.