Pancyclic Research Articles

Updating the hamiltonian problem—A survey

AbstractThis is intended as a survey article covering recent developments in the area of hamiltonian graphs, that is, graphs containing a spanning cycle. This article also contains some material on related topics such as traceable, hamiltonian‐connected and pancyclic graphs and digraphs, as well as an extensive bibliography of papers in the area.

“Panning”, a technique for enrichment of the oyster parasite Haplosporidium nelsoni (MSX)

Plasmodial stages of the parasite Haplosporidium nelsoni (MSX) were enriched an average eight-fold with recovery of 60 to 80% using a process known as “panning” of hemolymph from infected oysters ( Crassostrea virginica). The method takes advantage of the greater adherence of hemocytes, compared to parasites, to inert surfaces. Hemolymph from infected oysters was layered onto a Petri dish and cells were allowed to settle for 30 min. Nonadhering cells were gently washed, decanted, and placed in a second Petri dish. Maximum enrichment and recovery of parasites were obtained after three panning cycles. Viability, estimated by trypan blue dye exclusion, was greater than 95%. Parasite-to-hemocyte ratios before panning were highly correlated to those after panning, but correlation decreased with each incubation. Initial concentration of parasites, which ranged from 1.4 to 5 × 10 5/ml, was also significantly correlated with the percentage recovery. We suggest that other extracellular invertebrate protozoans may also be enriched by the same method, which is much simpler, gentler, and less expensive than most other cell separation procedures.

Some theorems of uniquely pancyclic graphs

In this paper we discuss an unsolved problem in [1]: Determine which simple graph G has exactly one cycle of each length l, 3⩽ l⩽ ν (where ν is the number of the vertices of G). We call a graph with this property a uniquely pancyclic graph (UPC-graph). We solve this problem under the condition: G is an outerplanar graph. We determine all UPC-graphs each of which contains ν + m edges for m⩽3. We also conjecture that none of the graphs, each of which contains ν + m edges for m⩾4, is a UPC-graph, and we prove that this conjecture for m=4 is true.

A simple panning method for the selection of cell surface antigen transfectants

Mouse Ltk − (H-2 k) cells that became reactive with anti-H-2 d monoclonal antibodies (McAb) after transfection with DNA from H-2 d-positive cells were isolated by an indirect panning method. The cells were transfected with DNA from a plasmid containing the selective thymidine kinase (tk) gene in addition to total cell DNA, so that a first selection could be carried out in hypoxanthine/aminopterin/thymidine (HAT) medium. The HAT-selected tk + transfectants were incubated with anti-H-2 d McAb, washed and transferred to dishes coated with purified anti-immunoglobulin (anti-Ig). This 2-step method of panning has the advantage that only the anti-Ig reagent requires purification. Transfected cells clearly reactive with either anti-L d/R d or anti-K d McAb were isolated after only 1 or2 cycles of panning.

Orderly algorithms for generating restricted classes of graphs

AbstractOrderly algorithms for the generation of exhaustive lists of nonisomorphic graphs are discussed. The existence of orderly methods to generate the graphs with a given subgraph and without a given subgraph is established. This method can be used to list all the nonisomorphic subgraphs of a given graph, as well as to produce catalogs of Hamiltonian graphs, pancyclic graphs, degree‐constrained graphs, and other classes. A generalization of this method is given that can be used to generate lists of graphs with given girth, planar graphs, k‐colorable graphs, and k‐connected graphs, for example. Finally, these observations are employed to generate restricted classes of digraphs, notably acyclic digraphs and poset digraphs. The generation of poset digraphs is shown to supply a practical orderly method for producing a catalog of lattices. Similar observations concerning vertex addition generation methods allow one to improve on existing methods for the generation of catalog of interval and circle graphs.

Some 4-valent, 3-connected, planar, almost pancyclic graphs

For each positive integer k (≠ 5,6), a 4-valent, 3-connected, planar graph, having cycles of all (possible) lengths except k is constructed.

Pancyclic graphs I

A graph G with vertex set V( G) and edge set E( G) is pancyclic if it contains cycles of all lengths l, 3 ≤ l ≤ | V( G) |. Theorem. Let G be Hamiltonian and suppose that |E(G)| ≥ n 2 4 , where n = | V( G)|. Then G is either pancyclic or else is the complete bipartite graph K n 2 , n 2 . As a corollary to this theorem it is shown that the Ore conditions for a graph to be Hamiltonian actually imply that the graph is either pancyclic or else is K n 2 , n 2 .