Star colouring and locally constrained graph homomorphisms

Dvo\v{r}\'ak, Mohar and \v{S}\'amal (J. Graph Theory, 2013) proved that for every 3-regular graph $G$, the line graph of $G$ is 4-star colourable if and only if $G$ admits a locally bijective homomorphism to the cube $Q_3$. We generalise this result as follows: for $p\geq 2$, a $K_{1,p+1}$-free $2p$-regular graph $G$ admits a $(p + 2)$-star colouring if and only if $G$ admits a locally bijective homomorphism to a fixed $2p$-regular graph named $G_{2p}$. We also prove the following: (i) for $p\geq 2$, a $2p$-regular graph $G$ admits a $(p + 2)$-star colouring if and only if $G$ has an orientation $\vec{G}$ that admits an out-neighbourhood bijective homomorphism to a fixed orientation $\vec{G_{2p}}$ of $G2p$; (ii) for every 3-regular graph $G$, the line graph of $G$ is 4-star colourable if and only if $G$ is bipartite and distance-two 4-colourable; and (iii) it is NP-complete to check whether a planar 4-regular 3-connected graph is 4-star colourable

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 cm3 PTW 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

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

Indium catalysts for low-pressure CO2/epoxide ring-opening copolymerization: Evidence for a mononuclear mechanism?

The alternating copolymerization of CO2/epoxides is a useful means to incorporate high levels of carbon dioxide into polymers. The reaction is generally proposed to occur by bimetallic or bicomponent pathways. Here, the first indium catalysts are presented, which are proposed to operate by a distinct mononuclear pathway. The most active and selective catalysts are phosphasalen complexes, which feature ligands comprising two iminophosphoranes linked to sterically hindered ortho-phenolates. The catalysts are active at 1 bar pressure of carbon dioxide and are most effective without any cocatalyst. They show low-pressure activity (1 bar pressure) and yield polymer with high carbonate linkage selectivity (>99%) and isoselectivity ( Pm > 70%). Using these complexes, it is also possible to isolate and characterize key catalytic intermediates, including the propagating indium alkoxide and carbonate complexes that are rarely studied. The catalysts are mononuclear under polymerization conditions, and the key intermediates show different coordination geometries: the alkoxide complex is pentacoordinate, while the carbonate is hexacoordinate. Kinetic analyses reveal a first-order dependence on catalyst concentration and are zero-order in carbon dioxide pressure; these findings together with in situ spectroscopic studies underpin the mononuclear pathway. More generally, this research highlights the future opportunity for other homogeneous catalysts, featuring larger ionic radius metals and new ligands, to operate by mononuclear mechanisms

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