Optimal acyclic edge colouring of grid like graphs

AbstractWe determine the values of the acyclic chromatic index of a class of graphs referred to as d-dimensional partial tori. These are graphs which can be expressed as the cartesian product of d graphs each of which is an induced path or cycle. This class includes some known classes of graphs like d-dimensional meshes, hypercubes, tori, etc. Our estimates are exact except when the graph is a product of a path and a number of odd cycles, in which case the estimates differ by an additive factor of at most 1. Our results are also constructive and provide an optimal (or almost optimal) acyclic edge colouring in polynomial time

Paths of specified length in random k-partite graphs

Fix positive integers k and l. Consider a random k-partite graph on n vertices obtained by partitioning the vertex set into V_i, (i=1, \ldots,k) each having size Ω (n) and choosing each possible edge with probability p. Consider any vertex x in any V_i and any vertex y. We show that the expected number of simple paths of even length l between x and y differ significantly depending on whether y belongs to the same V_i (as x does) or not. A similar phenomenon occurs when l is odd. This result holds even when k,l vary slowly with n. This fact has implications to coloring random graphs. The proof is based on establishing bijections between sets of paths

On the size of induced acyclic subgraphs in random digraphs

Graphs and Algorithm

Graphs of low chordality

The chordality of a graph with at least one cycle is the length of the longest induced cycle in it. The odd (even) chordality is defined to be the length of the longest induced odd (even) cycle in it. Chordal graphs have chordality at most 3. We show that co-circular-arc graphs and co-circle graphs have even chordality at most 4. We also identify few other classes of graphs having bounded (by a constant) chordality values

Molecular Characterization of a Novel Intracellular ADP-Ribosyl Cyclase

Background. ADP-ribosyl cyclases are remarkable enzymes capable of catalyzing multiple reactions including the synthesis of the novel and potent intracellular calcium mobilizing messengers, cyclic ADP-ribose and NAADP. Not all ADP-ribosyl cyclases however have been characterized at the molecular level. Moreover, those that have are located predominately at the outer cell surface and thus away from their cytosolic substrates. Methodology/Principal Findings. Here we report the molecular cloning of a novel expanded family of ADP-ribosyl cyclases from the sea urchin, an extensively used model organism for the study of inositol trisphosphate-independent calcium mobilization. We provide evidence that one of the isoforms (SpARC1) is a soluble protein that is targeted exclusively to the endoplasmic reticulum lumen when heterologously expressed. Catalytic activity of the recombinant protein was readily demonstrable in crude cell homogenates, even under conditions where luminal continuity was maintained. Conclusions/Significance. Our data reveal a new intracellular location for ADP-ribosyl cyclases and suggest that production of calcium mobilizing messengers may be compartmentalized

Whole-genome sequencing reveals host factors underlying critical COVID-19

Critical COVID-19 is caused by immune-mediated inflammatory lung injury. Host genetic variation influences the development of illness requiring critical care1 or hospitalization2,3,4 after infection with SARS-CoV-2. The GenOMICC (Genetics of Mortality in Critical Care) study enables the comparison of genomes from individuals who are critically ill with those of population controls to find underlying disease mechanisms. Here we use whole-genome sequencing in 7,491 critically ill individuals compared with 48,400 controls to discover and replicate 23 independent variants that significantly predispose to critical COVID-19. We identify 16 new independent associations, including variants within genes that are involved in interferon signalling (IL10RB and PLSCR1), leucocyte differentiation (BCL11A) and blood-type antigen secretor status (FUT2). Using transcriptome-wide association and colocalization to infer the effect of gene expression on disease severity, we find evidence that implicates multiple genes—including reduced expression of a membrane flippase (ATP11A), and increased expression of a mucin (MUC1)—in critical disease. Mendelian randomization provides evidence in support of causal roles for myeloid cell adhesion molecules (SELE, ICAM5 and CD209) and the coagulation factor F8, all of which are potentially druggable targets. Our results are broadly consistent with a multi-component model of COVID-19 pathophysiology, in which at least two distinct mechanisms can predispose to life-threatening disease: failure to control viral replication; or an enhanced tendency towards pulmonary inflammation and intravascular coagulation. We show that comparison between cases of critical illness and population controls is highly efficient for the detection of therapeutically relevant mechanisms of disease

Paths of specified length in random k-partite graphs

Fix positive integers k and l. Consider a random k-partite graph on n vertices obtained by partitioning the vertex set into V_i, (i=1, \ldots,k) each having size Ω (n) and choosing each possible edge with probability p. Consider any vertex x in any V_i and any vertex y. We show that the expected number of simple paths of even length l between x and y differ significantly depending on whether y belongs to the same V_i (as x does) or not. A similar phenomenon occurs when l is odd. This result holds even when k,l vary slowly with n. This fact has implications to coloring random graphs. The proof is based on establishing bijections between sets of paths

Intersection dimension and graph invariants

We show that the intersection dimension of graphs with respect to several hereditary properties can be bounded as a function of the maximum degree. As an interesting special case, we show that the circular dimension of a graph with maximum degree Δis at most O(ΔlogΔ/log logΔ) . It is also shown that permutation dimension of any graph is at most Δ(log Δ)1+o(1). We also obtain bounds on intersection dimension in terms of treewidth

Intersection Dimension and Graph Invariants

We show that the intersection dimension of graphs with respect to several hereditary properties can be bounded as a function of the maximum degree. As an interesting special case, we show that the circular dimension of a graph with maximum degree Δ is at most O(ΔlogΔlog logΔ)O\left( {\Delta {{\log \Delta } \over {\log \,\log \Delta }}} \right) . It is also shown that permutation dimension of any graph is at most Δ(log Δ)1+o(1). We also obtain bounds on intersection dimension in terms of treewidth