A class of additive multiplicative graph functions

AbstractFor a fixed graph G, the capacity function for G, PG, is defined by PG(H) = limn→∞[γG(Hn)]1/n, where γG(H) is the maximum number of disjoint G's in H. In [2], Hsu proved that PK2 is multiplicative or not. In this paper, we prove that PG is multiplicative and additive for some graphs G which include K2. Some properties of PG are also discussed in this paper

Globally bi-3∗-connected graphs

AbstractA k-container C(x,y) in a graph G=(V,E) is a set of k internally node-disjoint paths between vertices x and y. A k∗-container C(x,y) of G is a k-container such that every vertex of G is incident with a certain path in C(x,y). A bipartite graph G=(B∪W,E) is globally bi-3∗-connected if there is a 3∗-container C(x,y) between any pair of vertices {x,y} with x∈B and y∈W. Furthermore, G is hyper globally bi-3∗-connected if it is globally bi-3∗-connected and there exists a 3∗-container C(x,y) in G−{z} for any three different vertices x,y, and z of the same partite set of G. A graph G=(V,E) is 1-edge Hamiltonian if G−e is Hamiltonian for any e∈E. A bipartite graph G=(B∪W,E) is 1p-Hamiltonian if G−{x,y} is Hamiltonian for any pair of vertices {x,y} with x∈B and y∈W. In this paper, we prove that every hyper globally bi-3∗-connected graph is 1p-Hamiltonian and every globally bi-3∗-connected graph is 1-edge Hamiltonian. We present some examples of hyper globally bi-3∗-connected graphs, some globally bi-3∗-connected graphs that are not hyper globally bi-3∗-connected, and some examples of 1-edge Hamiltonian bipartite graphs that are not globally bi-3∗-connected

On the extremal number of edges in hamiltonian connected graphs

AbstractAssume that n and δ are positive integers with 3≤δ<n. Let hc(n,δ) be the minimum number of edges required to guarantee an n-vertex graph G with minimum degree δ(G)≥δ to be hamiltonian connected. Any n-vertex graph G with δ(G)≥δ is hamiltonian connected if |E(G)|≥hc(n,δ). We prove that hc(n,δ)=C(n−δ+1,2)+δ2−δ+1 if δ≤⌊n+3×(nmod2)6⌋+1, hc(n,δ)=C(n−⌊n2⌋+1,2)+⌊n2⌋2−⌊n2⌋+1 if ⌊n+3×(nmod2)6⌋+1<δ≤⌊n2⌋, and hc(n,δ)=⌈nδ2⌉ if δ>⌊n2⌋

Conditional fault hamiltonian connectivity of the complete graph

A path in G is a hamiltonian path if it contains all vertices of G. A graph G is hamiltonian connected if there exists a hamiltonian path between any two distinct vertices of G. The degree of a vertex u in G is the number of vertices of G adjacent to u. We denote by is defined as the maximum integer k such that G is k edge-fault tolerant conditional hamiltonian connected if G is hamiltonian connected and is undefined otherwise. Let n 4. We use K n to denote the complete graph with n vertices. In this paper, we show tha

Fault-tolerant hamiltonian connectedness of cycle composition networks

Abstract It is important for a network to tolerate as many faults as possible. With the graph representation of an interconnection network, a k-regular hamiltonian and hamiltonian connected network is super fault-tolerant hamiltonian if it remains hamiltonian after removing up to k À 2 vertices and/or edges and remains hamiltonian connected after removing up to k À 3 vertices and/or edges. Super fault-tolerant hamiltonian networks have an optimal flavor with regard to the fault-tolerant hamiltonicity and fault-tolerant hamiltonian connectivity. For this reason, a cycle composition framework was proposed to construct a (k + 2)-regular super fault-tolerant hamiltonian network based on a collection of n k-regular super fault-tolerant hamiltonian networks containing the same number of vertices for n P 3 and k P 5. This paper is aimed to emphasize that the cycle composition framework can be still applied even when k = 4

Women with endometriosis have higher comorbidities: Analysis of domestic data in Taiwan

AbstractEndometriosis, defined by the presence of viable extrauterine endometrial glands and stroma, can grow or bleed cyclically, and possesses characteristics including a destructive, invasive, and metastatic nature. Since endometriosis may result in pelvic inflammation, adhesion, chronic pain, and infertility, and can progress to biologically malignant tumors, it is a long-term major health issue in women of reproductive age. In this review, we analyze the Taiwan domestic research addressing associations between endometriosis and other diseases. Concerning malignant tumors, we identified four studies on the links between endometriosis and ovarian cancer, one on breast cancer, two on endometrial cancer, one on colorectal cancer, and one on other malignancies, as well as one on associations between endometriosis and irritable bowel syndrome, one on links with migraine headache, three on links with pelvic inflammatory diseases, four on links with infertility, four on links with obesity, four on links with chronic liver disease, four on links with rheumatoid arthritis, four on links with chronic renal disease, five on links with diabetes mellitus, and five on links with cardiovascular diseases (hypertension, hyperlipidemia, etc.). The data available to date support that women with endometriosis might be at risk of some chronic illnesses and certain malignancies, although we consider the evidence for some comorbidities to be of low quality, for example, the association between colon cancer and adenomyosis/endometriosis. We still believe that the risk of comorbidity might be higher in women with endometriosis than that we supposed before. More research is needed to determine whether women with endometriosis are really at risk of these comorbidities

Guidelines for the use and interpretation of assays for monitoring autophagy (3rd edition)

In 2008 we published the first set of guidelines for standardizing research in autophagy. Since then, research on this topic has continued to accelerate, and many new scientists have entered the field. Our knowledge base and relevant new technologies have also been expanding. Accordingly, it is important to update these guidelines for monitoring autophagy in different organisms. Various reviews have described the range of assays that have been used for this purpose. Nevertheless, there continues to be confusion regarding acceptable methods to measure autophagy, especially in multicellular eukaryotes. For example, a key point that needs to be emphasized is that there is a difference between measurements that monitor the numbers or volume of autophagic elements (e.g., autophagosomes or autolysosomes) at any stage of the autophagic process versus those that measure fl ux through the autophagy pathway (i.e., the complete process including the amount and rate of cargo sequestered and degraded). In particular, a block in macroautophagy that results in autophagosome accumulation must be differentiated from stimuli that increase autophagic activity, defi ned as increased autophagy induction coupled with increased delivery to, and degradation within, lysosomes (inmost higher eukaryotes and some protists such as Dictyostelium ) or the vacuole (in plants and fungi). In other words, it is especially important that investigators new to the fi eld understand that the appearance of more autophagosomes does not necessarily equate with more autophagy. In fact, in many cases, autophagosomes accumulate because of a block in trafficking to lysosomes without a concomitant change in autophagosome biogenesis, whereas an increase in autolysosomes may reflect a reduction in degradative activity. It is worth emphasizing here that lysosomal digestion is a stage of autophagy and evaluating its competence is a crucial part of the evaluation of autophagic flux, or complete autophagy. Here, we present a set of guidelines for the selection and interpretation of methods for use by investigators who aim to examine macroautophagy and related processes, as well as for reviewers who need to provide realistic and reasonable critiques of papers that are focused on these processes. These guidelines are not meant to be a formulaic set of rules, because the appropriate assays depend in part on the question being asked and the system being used. In addition, we emphasize that no individual assay is guaranteed to be the most appropriate one in every situation, and we strongly recommend the use of multiple assays to monitor autophagy. Along these lines, because of the potential for pleiotropic effects due to blocking autophagy through genetic manipulation it is imperative to delete or knock down more than one autophagy-related gene. In addition, some individual Atg proteins, or groups of proteins, are involved in other cellular pathways so not all Atg proteins can be used as a specific marker for an autophagic process. In these guidelines, we consider these various methods of assessing autophagy and what information can, or cannot, be obtained from them. Finally, by discussing the merits and limits of particular autophagy assays, we hope to encourage technical innovation in the field

Optimal k-Fault-tolerant . . .

... In this paper, we discuss the case of a combination of processor failures and link failures for token rings. By &quot;k faults&quot; we mean k-component faults in any combination of processor faults and link faults. Designing an optimal k-fault- tolerant network for token rings is equivalent to constructing an optimal k-hamiltonian graph, where k is a positive integer and corresponds to the number of faults. A graph G is k-hamiltonian if G - F is hamiltonian for any sets F V E with |F| k. An n- node k-hamiltonian graph is optimal if it contains the least number of edges among all n-node k-hamiltonian graphs. In this paper, we construct optimal k-hamiltonian graphs with k = 2 and 3, which are optimal k-fault-tolerant networks with respect to token ring