Hoede-sequences

In an attempt to prove the double-cycle-conjecture for cubic graphs,\ud C. Hoede formulated the following combinatorial problem.\ud “Given a partition of {1, 2, . . . , 3n} into n equal classes, is\ud it possible to choose from each class a number such that\ud these numbers form an increasing sequence of alternating\ud parity?U+00e2U+0080?\ud Let a Hoede-sequence be defined as an increasing sequence of natural\ud numbers of alternating parity. We determine the average number of\ud Hoede-sequences w.r.t. arbitrary partitions, and obtain bounds for the\ud maximum and minimum number of Hoede-sequences w.r.t. partitions\ud into equal classes.\u

Even graphs

A nontrivial connected graph G is called even if for each vertex v of G there is a unique vertex [bar v] such that d(v,[bar v]) = diam G. Special classes of even graphs are defined and compared to each other. In particular, an even graph G is called symmetric if d(u,v) + d(u,[bar v]) = diam G for all u, v V(G). Several properties of even and symmetric even graphs are stated. For an even graph of order n and diameter d other than an even cycle it is shown that n ≥ 3d - 1 and conjectured that n ≥ 4d - 4. This conjecture is proved for symmetric even graphs and it is shown that for each pair of integers n, d with n even, d ≥ 2 and n ≥ 4d - 4 there exists an even graph of order n and diameter d. Several ways of constructing new even graphs from known ones are presented

Z-related pairs in microtonal systems

Various infinite families of Z-related pairs in microtonal systems are presented. Soderberg's dual inversion is compared to a more special transformation, the one-pitch shift. The material is illustrated by several examples. \u

Coloring a graph optimally with two colors

AbstractLet G be a graph with point set V. A (2-)coloring of G is a map of V to {red, white}. An error occurs whenever the two endpoints of a line have the same color. An optimal coloring of G is a coloring of G for which the number of errors is minimum. The minimum number of errors is denoted by γ(G), we derive upper and lower bounds for γ(G) and prove that if G is a graph with n points and m lines, then max{0, m−⌊14n2⌋}⩽γ(G)⩽⌊12m−14(h(m)−1)⌋, where h(m)=min{n¦m⩽(n2)}. The lower bound is sharp, and for infinitely many values of m the upper bound is attained for all sufficiently large n

On an optimality property of ternary trees

The concept of effort is defined for rooted trees. The class of rooted trees with minimal effort is determined. The asymptotic behaviour of the minimal effort is calculated. Various choices for the effort function are considered, as well as variations of the optimality criterion

Graph theoretic aspects of music theory

The cycle on twelve points is a well-known representation of the twelve pitch classes of the traditional scale. We treat a more general situation where the number of pitch classes can be different from twelve and where, moreover, other measures of closeness are taken into account. We determine all situations for which the Generalized Hexachord Theorem continues to hold. \u

Label-connected graphs and the gossip problem

A graph with m edges is called label-connected if the edges can be labeled with real numbers in such a way that, for every pair (u, v) of vertices, there is a (u, v)-path with ascending labels. The minimum number of edges of a label-connected graph on n vertices equals the minimum number of calls in the gossip problem for n persons, which is known to be 2n − 4 for n ≥ 4. A polynomial characterization of label-connected graphs with n vertices and 2n − 4 edges is obtained. For a graph G, let θ(G) denote the minimum number of edges that have to be added to E(G) in order to create a graph with two edge-disjoint spanning trees. It is shown that for a graph G to be label-connected, θ(G) ≤ 2 is necessary and θ(G) ≤ 1 is sufficient. For i = 1, 2, the condition θ(G) ≤ i can be checked in polynomial time. Yet recognizing label-connected graphs is an NP-complete problem. This is established by first showing that the following problem is NP-complete: Given a graph G and two vertices u and v of G, does there exist a (u, v)-path P in G such that G−E(P) is connected

NK and T cells constitute two major, functionally distinct intestinal epithelial lymphocyte subsets in the chicken

Non-mammalian NK cells have not been characterized in detail; however, their analysis is essential for the understanding of the NK cell receptor phylogeny. As a first step towards defining chicken NK cells, several tissues were screened for the presence of NK cells, phenotypically defined as CD8(+) cells lacking T- or B-lineage specific markers. By this criteria, approximately 30% of CD8(+) intestinal intraepithelial lymphocytes (IEL), but <1% of splenocytes or peripheral blood lymphocytes were defined as NK cells. These CD8(+)CD3(-) IEL were used for the generation of the 28-4 mAb, immunoprecipitating a 35-kDa glycoprotein with a 28-kDa protein core. The CD3 and 28-4 mAb were used to separate IEL into CD3(+) IEL T cells and 28-4(+) cells, both co-expressing the CD8 antigen. During ontogeny, 28-4(+) cells were abundant in the IEL and in the embryonic spleen, where two subsets could be distinguished according to their CD8 and c-kit expression. Most importantly, 28-4(+) IEL lysed NK-sensitive targets, whereas intestinal T cells did not have any spontaneous cytolytic activity. These results define two major, phenotypically and functionally distinct IEL subpopulations, and imply an important role of NK cells in the mucosal immune system