Let S be any set of points in the Euclidean plane R2. For any p = (x, y) ∈ S, put SW (p) = {(x, y) ∈ S : x < x and y < y} and NE(p) = {(x, y) ∈ S : x > x and y > y}. Let GS be the graph with vertex set S and edge set {pq : NE(p) ∩ NE(q) 6= ∅ and SW (p) ∩ SW (q) 6= ∅}. We prove that the graphH with V (H) = {u, v, z, w, p, p1, p2, p3} and E(H) = {uv, vz, zw, wu, p1p3, p2p3, pu, pv, pz, pw, pp1, p...

An (n, a, b)-perfect double cube is a b × b × b sized n-ary periodic array containing all possible a × a × a sized n-ary array exactly once as subarray. A growing cube is an array whose cj × cj × cj sized prefix is an (nj, a, cj)-perfect double cube for j = 1, 2, . . ., where cj = n v/3 j , v = a 3 and n1 < n2 < · · ·. We construct the smallest possible perfect double cube (a 256×256×256 sized ...

Continuing work initiated by Häggkvist and Markström, we show in this paper that certain disconnected frames guarantee the existence of a cycle double cover. Specifically, we show that the disjoint union of a Kotzig and a sturdy graph forms a good frame.

The maximum number of non-crossing straight-line perfect matchings that a set of n points in the plane can have is known to be O(10.0438) and Ω∗(3n). The lower bound, due to Garćıa, Noy, and Tejel (2000), is attained by the double chain, which has Θ(3/n) such matchings. We reprove this bound in a simplified way that uses the novel notion of down-free matchings. We then apply this approach to se...

Define a graph to be a Kotzig graph if it is m-regular and has an m-edge colouring in which each pair of colours form a Hamiltonian cycle. We show that every cubic graph with spanning subgraph consisting of a subdivision of a Kotzig graph together with even cycles has a cycle double cover, in fact a 6-CDC. We prove this for two other families of graphs similar to Kotzig graphs as well. In parti...

Let ρn(V ) be the number of complete hyperbolic manifolds of dimension n with volume less than V . Burger, Gelander, Lubotzky, and Moses[2] showed that when n ≥ 4 there exist a, b > 0 depending on the dimension such that aV logV ≤ log ρn(V ) ≤ bV logV, for V ≫ 0. In this note, we use their methods to bound the number of hyperbolic manifolds with diameter less than d and show that the number gro...

A cycle C in a graph is called stable if there exist no other cycle D in the same graph such that V (C) ⊆ V (D). In this paper we study stable cycles in snarks and based on our findings we are able to show that if a cubic graph G has a cycle of length at least |V (G)| − 9 then it has a cycle double cover. We also give a construction for an infinite snark family with stable cycles of constant le...

We define a general scheme for the evolution of fragmentation functions which resums soft gluon logarithms in a manner consistent with fixed order evolution. We present an explicit example of our approach in which double logarithms are resummed using the Double Logarithmic Approximation. We show that this scheme reproduces the Modified Leading Logarithm Approximation in certain limits, and find...

This paper deals with (global) finite-gain input/output stabilization of linear systems with saturated controls. For neutrally stable systems, it is shown that the linear feedback law suggested by the passivity approach indeed provides stability, with respect to every L-norm. Explicit bounds on closed-loop gains are obtained, and they are related to the norms for the respective systems without ...

A semiextension of a circuit C in a graph G provides a possible means of finding a cycle double cover of G with C as a prescribed circuit. Recently we conjectured [E.E. García Moreno, T.R. Jensen, On semiextensions and circuit double covers, J. Combin. Theory Ser. B 97 (2007) 474–482] that if G is cubic and 2-edge-connected, then a semiextension of C in G exists. If true, this would imply sever...