Star Colouring of Bounded Degree Graphs and Regular Graphs

A $k$-star colouring of a graph $G$ is a function $f:V(G)\to\{0,1,\dots,k-1\}$ such that $f(u)\neq f(v)$ for every edge $uv$ of $G$, and every bicoloured connected subgraph of $G$ is a star. The star chromatic number of $G$, $\chi_s(G)$, is the least integer $k$ such that $G$ is $k$-star colourable. We prove that $\chi_s(G)\geq \lceil (d+4)/2\rceil$ for every $d$-regular graph $G$ with $d\geq 3$. We reveal the structure and properties of even-degree regular graphs $G$ that attain this lower bound. The structure of such graphs $G$ is linked with a certain type of Eulerian orientations of $G$. Moreover, this structure can be expressed in the LC-VSP framework of Telle and Proskurowski (SIDMA, 1997), and hence can be tested by an FPT algorithm with the parameter either treewidth, cliquewidth, or rankwidth. We prove that for $p\geq 2$, a $2p$-regular graph $G$ is $(p+2)$-star colourable only if $n:=|V(G)|$ is divisible by $(p+1)(p+2)$. For each $p\geq 2$ and $n$ divisible by $(p+1)(p+2)$, we construct a $2p$-regular Hamiltonian graph on $n$ vertices which is $(p+2)$-star colourable. The problem $k$-STAR COLOURABILITY takes a graph $G$ as input and asks whether $G$ is $k$-star colourable. We prove that 3-STAR COLOURABILITY is NP-complete for planar bipartite graphs of maximum degree three and arbitrarily large girth. Besides, it is coNP-hard to test whether a bipartite graph of maximum degree eight has a unique 3-star colouring up to colour swaps. For $k\geq 3$, $k$-STAR COLOURABILITY of bipartite graphs of maximum degree $k$ is NP-complete, and does not even admit a $2^{o(n)}$-time algorithm unless ETH fails

Hardness Transitions of Star Colouring and Restricted Star Colouring

We study how the complexity of the graph colouring problems star colouring and restricted star colouring vary with the maximum degree of the graph. Restricted star colouring (in short, rs colouring) is a variant of star colouring. For $k\in \mathbb{N}$, a $k$-colouring of a graph $G$ is a function $f\colon V(G)\to \mathbb{Z}_k$ such that $f(u)\neq f(v)$ for every edge $uv$ of $G$. A $k$-colouring of $G$ is called a $k$-star colouring of $G$ if there is no path $u,v,w,x$ in $G$ with $f(u)=f(w)$ and $f(v)=f(x)$. A $k$-colouring of $G$ is called a $k$-rs colouring of $G$ if there is no path $u,v,w$ in $G$ with $f(v)>f(u)=f(w)$. For $k\in \mathbb{N}$, the problem $k$-STAR COLOURABILITY takes a graph $G$ as input and asks whether $G$ admits a $k$-star colouring. The problem $k$-RS COLOURABILITY is defined similarly. Recently, Brause et al. (Electron. J. Comb., 2022) investigated the complexity of 3-star colouring with respect to the graph diameter. We study the complexity of $k$-star colouring and $k$-rs colouring with respect to the maximum degree for all $k\geq 3$. For $k\geq 3$, let us denote the least integer $d$ such that $k$-STAR COLOURABILITY (resp. $k$-RS COLOURABILITY) is NP-complete for graphs of maximum degree $d$ by $L_s^{(k)}$ (resp. $L_{rs}^{(k)}$). We prove that for $k=5$ and $k\geq 7$, $k$-STAR COLOURABILITY is NP-complete for graphs of maximum degree $k-1$. We also show that $4$-RS COLOURABILITY is NP-complete for planar 3-regular graphs of girth 5 and $k$-RS COLOURABILITY is NP-complete for triangle-free graphs of maximum degree $k-1$ for $k\geq 5$. Using these results, we prove the following: (i) for $k\geq 4$ and $d\leq k-1$, $k$-STAR COLOURABILITY is NP-complete for $d$-regular graphs if and only if $d\geq L_s^{(k)}$; and (ii) for $k\geq 4$, $k$-RS COLOURABILITY is NP-complete for $d$-regular graphs if and only if $L_{rs}^{(k)}\leq d\leq k-1$

FLUKA Monte Carlo for Basic Dosimetric Studies of Dual Energy Medical Linear Accelerator

General purpose Monte Carlo code for simulation of particle transport is used to study the basic dosimetric parameters like percentage depth dose and dose profiles and compared with the experimental measurements from commercial dual energy medical linear accelerator. Varian Clinac iX medical linear accelerator with dual energy photon beams (6 and 15 MV) is simulated using FLUKA. FLAIR is used to visualize and edit the geometry. Experimental measurements are taken for 100 cm source-to-surface (SSD) in 50 × 50 × 50 cm3PTW water phantom using 0.12 cc cylindrical ionization chamber. Percentage depth dose for standard square field sizes and dose profiles for various depths are studied in detail. The analysis was carried out using ROOT (a DATA analysis frame work developed at CERN) system. Simulation result shows good agreement in percentage depth dose and beam profiles with the experimental measurements for Varian Clinac iX dual energy medical linear accelerator

Transcriptomic responses of the heart and brain to anoxia in the Western Painted turtle

Painted turtles are the most anoxia-tolerant tetrapods known, capable of surviving without oxygen for more than four months at 3°C and 30 hours at 20°C. To investigate the transcriptomic basis of this ability, we used RNA-seq to quantify mRNA expression in the painted turtle ventricle and telencephalon after 24 hours of anoxia at 19°C. Reads were obtained from 22,174 different genes, 13,236 of which were compared statistically between treatments for each tissue. Total tissue RNA contents decreased by 16% in telencephalon and 53% in ventricle. The telencephalon and ventricle showed ≥ 2x expression (increased expression) in 19 and 23 genes, respectively, while only four genes in ventricle showed ≤ 0.5x changes (decreased expression). When treatment effects were compared between anoxic and normoxic conditions in the two tissue types, 31 genes were increased (≥ 2x change) and 2 were decreased (≤ 0.5x change). Most of the effected genes were immediate early genes and transcription factors that regulate cellular growth and development; changes that would seem to promote transcriptional, translational, and metabolic arrest. No genes related to ion channels, synaptic transmission, cardiac contractility or excitation-contraction coupling changed. The generalized expression pattern in telencephalon and across tissues, but not in ventricle, correlated with the predicted metabolic cost of transcription, with the shortest genes and those with the fewest exons showing the largest increases in expression

Knowledge, Attitude and Practice towards Self-Medications in a Rural Community

Background: The consumption of drugs without consulting healthcare professional regardless of side effects and duration of therapy is known as self-medication. This study was conducted to analyze the practice of self-medication and knowledge about the medicines among the common people of rural areas. Methods:  6 months Study was carried out by collecting the data with the help of questionnaire and thereby collecting their answers. The data was analyzed statistically in terms of actual numbers and percentage. Results:  As an outcome of our survey, majority of the population were unaware about proper usage of medicines. The survey also shows that NSAIDs are the most consumed medicines. 72.11% stop taking medications without the advice or without informing healthcare provider. 77.56% believe medication without prescription will help them. Conclusion: The survey that most of the common population in the rural areas don’t know about the indication for which they are taking medication and the study also emphasizes  the unawareness of common population towards proper usage of medication. Elderly people are on polypharmacy due to multiple and inter current diseases, with their improper use due to lack of knowledge of correct dose, side effects, and interactions would bring serious implications. Community pharmacists should get continuous education and repeated training programs because they are easy accessible to the common population. A proper statutory drug control must be implemented, rationally restricting the availability of drugs to the public. Self-medication is an area where governments and health authorities need to ensure that it is done in a responsible manner, ensuring that safe drugs are made available over the counter and the consumer is given adequate information about the use of drugs and when to consult a doctor. Keywords: Self-medication, irrational use, prevalence, pharmacist, drugs

dosimetric studies of mixed energy intensity modulated radiation therapy for prostate cancer treatments

Dosimetric studies of mixed field photon beam intensity modulated radiation therapy (IMRT) for prostate cancer using pencil beam (PB) and collapsed cone convolution (CCC) algorithms using Oncentra MasterPlan treatment planning system (v. 4.3) are investigated in this study. Three different plans were generated using 6 MV, 15 MV, and mixed beam (both 6 and 15 MV). Fifteen patients with two sets of plans were generated: one by using PB and the other by using CCC for the same planning parameters and constraints except the beam energy. For each patient's plan of high energy photons, one set of photoneutron measurements using solid state neutron track detector (SSNTD) was taken for this study. Mean percentage ofV66 Gyin the rectum is18.55±2.8,14.58±2.1, and16.77±4.7for 6 MV, 15 MV, and mixed-energy plans, respectively. Mean percentage ofV66 Gyin bladder is16.54±2.1,17.42±2.1,and16.94±41.9for 6 MV, 15 MV, and mixed-energy plans, respectively. Mixed fields neutron contribution at the beam entrance surface is 45.62% less than at 15 MV photon beam. Our result shows that, with negligible neutron contributions, mixed field IMRT has considerable dosimetric advantage