Abstract A connected graph $G$ is called strongly Menger edge connected if $G$ has min{deg$_{G}(x)$, deg$_{G}(y)$} edge-disjoint paths between any two distinct vertices $x$ and $y$ in $G$. In this paper, we consider two types of the strongly Menger edge connectivity of the line graphs of generalized $n$-dimensional hypercubes with faulty edges, namely the $m$-edge-fault-tolerant and $m$-conditional edge-fault-tolerant strongly Menger edge connectivity. We show that the line graphs of all generalized $n$-dimensional hypercubes are $(2n-4)$-edge-fault-tolerant strongly Menger edge connected for $n\geq 3$ and $(4n-10)$-conditional edge-fault-tolerant strongly Menger edge connected for $n\geq 4$. The two bounds for the maximum numbers of faulty edges are best possible.

Whitney proved in 1931 that every 4-connected planar triangulation is hamiltonian. Later, Hakimi, Schmeichel, and Thomassen in 1979 conjectured that every such triangulation on $n$ vertices has at least $2(n - 2)(n - 4)$ hamiltonian cycles. Along this direction, Brinkmann, Souffriau, and Van Cleemput in 2018 established a linear lower bound on the number of hamiltonian cycles in 4-connected planar triangulations. In stark contrast, Alahmadi, Aldred, and Thomassen in 2020 showed that every 5-connected triangulation of the plane or the projective plane has exponentially many hamiltonian cycles. This gives the motivation to study the number of hamiltonian cycles of 4-connected triangulations with few 4-separators. Recently, Liu and Yu in 2021 showed that every 4-connected planar triangulation with $O(n / \log n)$ 4-separators has a quadratic number of hamiltonian cycles. By adapting the framework of Alahmadi, Aldred, and Thomassen, we strengthen the last two aforementioned results. We prove that every 4-connected planar or projective planar triangulation with $O(n)$ 4-separators has exponentially many hamiltonian cycles.

Given a graph G and two non-negative integers p and q, an L(p, q)-labeling c of G is an assignment of non-negative integers to the vertices of G such that, for any two vertices u and v, |c(u) − c(v)| ≥ p if d(u, v)=1 and |c(u) − c(v)| ≥ q if d(u, v)=2. The L(p, q)-labeling problem arises from a variation of the channel assignment problem. We establish a connection between the L(p, q)-labeling of the planar graph G and the integer flow on the dual graph of G. This provides us with an alternative and potentially more effective way to minimize the edge span of L(p, q)-labelings for planar graphs by using a graph flow approach. As examples, we apply this approach to determine the minimum edge spans of the L(p, q)-labelings for some typical finite planar lattices, including the square lattice, triangular lattice, hexagonal lattice as well as some other lattices consisting of various regular polygons, some of which arise from the design of planar regions for cellular phone networks. Our flow-based approach is also expected to have some further applications in optimizing the other measures for the L(p, q)-labeling problem.

Simultaneous imaging of multiple probes or biomarkers represents a critical step toward high specificity molecular imaging. In this work, we propose to utilize the element-specific nature of the X-ray fluorescence (XRF) signal for imaging multiple elements simultaneously (multiplexing) using XRF computed tomography (XFCT). A 5-mm-diameter pencil beam produced by a polychromatic X-ray source (150 kV, 20 mA) was used to stimulate emission of XRF photons from 2% (weight/volume) gold (Au), gadolinium (Gd), and barium (Ba) embedded within a water phantom. The phantom was translated and rotated relative to the stationary pencil beam in a first-generation CT geometry. The X-ray energy spectrum was collected for 18 s at each position using a cadmium telluride detector. The spectra were then used to isolate the K shell XRF peak and to generate sinograms for the three elements of interest. The distribution and concentration of the three elements were reconstructed with the iterative maximum likelihood expectation maximization algorithm. The linearity between the XFCT intensity and the concentrations of elements of interest was investigated. We found that measured XRF spectra showed sharp peaks characteristic of Au, Gd, and Ba. The narrow full-width at half-maximum (FWHM) of the peaks strongly supports the potential of XFCT for multiplexed imaging of Au, Gd, and Ba ( FWHM(Au,Kα1) = 0.619 keV, FWHM(Au,Kα2)=1.371 keV , FWHM(Gd,Kα)=1.297 keV, FWHM(Gd,Kβ)=0.974 keV , FWHM(Ba,Kα)=0.852 keV, and FWHM(Ba,Kβ)=0.594 keV ). The distribution of Au, Gd, and Ba in the water phantom was clearly identifiable in the reconstructed XRF images. Our results showed linear relationships between the XRF intensity of each tested element and their concentrations ( R(2)(Au)=0.944 , R(Gd)(2)=0.986, and R(Ba)(2)=0.999), suggesting that XFCT is capable of quantitative imaging. Finally, a transmission CT image was obtained to show the potential of the approach for providing attenuation correction and morphological information. In conclusion, XFCT is a promising modality for multiplexed imaging of high atomic number probes.