Paired 2-disjoint path covers of burnt pancake graphs with faulty elements

The burnt pancake graph $BP_n$ is the Cayley graph of the hyperoctahedral group using prefix reversals as generators. Let $\{u,v\}$ and $\{x,y\}$ be any two pairs of distinct vertices of $BP_n$ for $n\geq 4$. We show that there are $u-v$ and $x-y$ paths whose vertices partition the vertex set of $BP_n$ even if $BP_n$ has up to $n-4$ faulty elements. On the other hand, for every $n\ge3$ there is a set of $n-2$ faulty edges or faulty vertices for which such a fault-free disjoint path cover does not exist.Comment: 14 pages, 4 figure

Mitochondrial function in Parkinson's disease and other neurodegenerative diseases

The cause of neuronal loss in the substantia nigra from Parkinson's disease (PD) brain is unknown. A deficiency of mitochondrial respiratory chain (MRC) complex I has been found in PD substantia nigra. Normal complex I function in multiple system atrophy suggested neither neuronal degeneration nor L-dopa therapy was the cause of this defect in PD. Normal MRC function in other specific brain areas from Alzheimer's disease (AD), dementia with Lewy bodies (DLB) and PD suggested that neither cholinergic cell loss nor the presence of Lewy bodies perse was associated with complex I deficiency. The increased ApoE ϵ4 allele frequency in AD and DLB was not observed in PD patients with dementia. These results suggested that the ApoE ϵ4 allele influenced neither the development of Lewy bodies nor the dementia associated with PD, and that the risk factors for dementia in PD differed from that of AD and DLB at least with respect to ApoE. In Huntington's disease (HD) caudate nucleus, severe defects of complexes II, III and complex IV activities of the MRC were demonstrated, supporting the role of abnormal energy metabolism in HD. The feasibility of the platelet-A549ρ° cell fusion technique to study the involvement of mitochondrial DNA (mtDNA) in PD was demonstrated using platelets with the A3243G mtDNA mutation (MELAS). Platelets from seven PD patients with low complex I activity were fused with A549ρ° cells. Mixed cybrid analysis demonstrated a selective 25% deficiency of complex I activity. Furthermore, the analysis of 16 A549ρ°-PD fusion cybrid clones from one of the patients expressed complexes I (25%) and IV (20%) deficiencies. These results point to abnormal mtDNA as the underlying cause of the MRC dysfunction in at least a proportion of PD patients. The MRC inhibitors piericidin A and antimycin A induced much greater levels of apoptosis in A549 than in A549ρ° cells implying apoptosis was induced via a mechanism that involved inhibition of the MRC, this contrasted with the toxic effects of rotenone which affected both cell types equally and therefore must be mediated via a pathway independent of the MRC

Fault diagnosability of regular graphs

An interconnection network\u27s diagnosability is an important measure of its self-diagnostic capability. In 2012, Peng et al. proposed a measure for fault diagnosis of the network, namely, the $h$-good-neighbor conditional diagnosability, which requires that every fault-free node has at least $h$ fault-free neighbors. There are two well-known diagnostic models, PMC model and MM* model. The {\it $h$-good-neighbor diagnosability} under the PMC (resp. MM*) model of a graph $G$, denoted by $t_h^{PMC}(G)$ (resp. $t_h^{MM^*}(G)$), is the maximum value of $t$ such that $G$ is $h$-good-neighbor $t$-diagnosable under the PMC (resp. MM*) model. In this paper, we study the $2$-good-neighbor diagnosability of some general $k$-regular $k$-connected graphs $G$ under the PMC model and the MM* model. The main result $t_2^{PMC}(G)=t_2^{MM^*}(G)=g(k-1)-1$ with some acceptable conditions is obtained, where $g$ is the girth of $G$. Furthermore, the following new results under the two models are obtained: $t_2^{PMC}(HS_n)=t_2^{MM^*}(HS_n)=4n-5$ for the hierarchical star network $HS_n$, $t_2^{PMC}(S_n^2)=t_2^{MM^*}(S_n^2)=6n-13$ for the split-star networks $S_n^2$ and $t_2^{PMC}(\Gamma_{n}(\Delta))=t_2^{MM^*}(\Gamma_{n}(\Delta))=6n-16$ for the Cayley graph generated by the $2$-tree $\Gamma_{n}(\Delta)$