Near Optimal Colourability on Hereditary Graph Families

A graph family $\mathcal{G}$ is near optimal colourable if there is a constant number $c$ such that every graph $G\in\mathcal{G}$ satisfies $\chi(G)\leq\max\{c,\omega(G)\}$, where $\chi(G)$ and $\omega(G)$ are the chromatic number and clique number of $G$, respectively. The near optimal colourable graph families together with the Lov{\'a}sz theta function are useful for the study of the chromatic number problems for hereditary graph families. In this paper, we investigate the near optimal colourability for ($H_1,H_2$)-free graphs. Our main result is an almost complete characterization for the near optimal colourability for ($H_1,H_2$)-free graphs with two exceptional cases, one of which is the celebrated Gy{\'a}rf{\'a}s conjecture. To obtain the result, we prove that the family of ($2K_2,P_4\vee K_n$)-free graphs is near optimal colourable for every positive integer $n$ by inductive arguments.Comment: 11 pages, 1 figur

Colouring graphs with no induced six-vertex path or diamond

The diamond is the graph obtained by removing an edge from the complete graph on 4 vertices. A graph is ($P_6$, diamond)-free if it contains no induced subgraph isomorphic to a six-vertex path or a diamond. In this paper we show that the chromatic number of a ($P_6$, diamond)-free graph $G$ is no larger than the maximum of 6 and the clique number of $G$. We do this by reducing the problem to imperfect ($P_6$, diamond)-free graphs via the Strong Perfect Graph Theorem, dividing the imperfect graphs into several cases, and giving a proper colouring for each case. We also show that there is exactly one 6-vertex-critical ($P_6$, diamond, $K_6$)-free graph. Together with the Lov\'asz theta function, this gives a polynomial time algorithm to compute the chromatic number of ($P_6$, diamond)-free graphs.Comment: 29 page

Colouring graphs with no induced six-vertex path or diamond

The diamond is the graph obtained by removing an edge from the complete graph on 4 vertices. A graph is (P6, diamond)-free if it contains no induced subgraph isomorphic to a six-vertex path or a diamond. In this paper we show that the chromatic number of a (P6, diamond)-free graph G is no larger than the maximum of 6 and the clique number of G. We do this by reducing the problem to imperfect (P6, diamond)-free graphs via the Strong Perfect Graph Theorem, dividing the imperfect graphs into several cases, and giving a proper colouring for each case. We also show that there is exactly one 6-vertex -critical (P6, diamond, K6)-free graph. Together with the Lovasz theta function, this gives a polynomial time algorithm to compute the chromatic number of (P6, diamond)-free graphs.(c) 2022 Elsevier B.V. All rights reserved

Colouring graphs with no induced six-vertex path or diamond

The diamond is the graph obtained by removing an edge from the complete graph on 4 vertices. A graph is (P-6, diamond)-free if it contains no induced subgraph isomorphic to a six-vertex path or a diamond. In this paper we show that the chromatic number of a (P-6, diamond)-free graph G is no larger than the maximum of 6 and the clique number of G. We do this by reducing the problem to imperfect (P-6, diamond)-free graphs via the Strong Perfect Graph Theorem, dividing the imperfect graphs into several cases, and giving a proper colouring for each case. We also show that there is exactly one 6-vertex-critical (P-6, diamond, K-6)-free graph. Together with the Lovasz theta function, this gives a polynomial time algorithm to compute the chromatic number of (P-6, diamond)-free graphs

Short-term Sulpiride Treatment of Children and Adolescents With Tourette Syndrome or Chronic Tic Disorder

Tourette syndrome (TS) is characterized by motor and vocal tics, and its diagnosis is based on clinical criteria. Dopamine-blocking neuroleptics are regarded as the most effective drugs for the treatment of TS. Sulpiride is a selective dopamine D2 antagonist. However, only one study with a large number of patients has reported the effect of treatment of TS with sulpiride. The purpose of this study was to evaluate prospectively the effect of sulpiride treatment of children and adolescents with TS or chronic tic disorder. Methods: The inclusion criteria were patients who fulfilled the diagnosis of TS or chronic tic disorder, and who had not received previous treatment. The severity of TS was assessed by the Yale Global Tic Severity Score (YGTSS) every 2 weeks for a total of 6 weeks. The patients started treatment with low-dose sulpiride according to their age on the first visit. The adverse effects of sulpiride were evaluated by subjective complaints from the patients themselves or their parents. The change in scores between each assessment point was analyzed by repeated measures one-way analysis of variance, with SPSS version 12.0 software. Results: One hundred and eighty-nine patients were enrolled. Their average age was 8.0 ± 2.5 years (range, 3–15 years). Most patients were male (n = 165, 87.3%). Six weeks' treatment significantly improved motor tics (p < 0.05), vocal tics (p < 0.05) and total YGTSS (p < 0.05). The most commonly encountered adverse effect was sedation (n = 31, 16.4%). Conclusion: Sulpiride is effective for short-term treatment of children and adolescents with TS or chronic tic disorder, and has few adverse effects

Endoplasmic Reticulum Stress in Elderly Patients with COVID-19: Potential of Melatonin Treatment

Aging processes, including immunosenescence, inflammation, inflammasome formation, genomic instability, telomeric attrition, and altered autophagy, are involved in viral infections and they may contribute to increased pathophysiological responses to the SARS-CoV-2 infection in the elderly; this poses additional risks of accelerated aging, which could be found even after recovery. Aging is associated with oxidative damage. Moreover, SARS-CoV-2 infections may increase the production of reactive oxygen species and such infections will disturb the Ca++ balance via an endoplasmic reticulum (ER) stress-mediated unfolded protein response. Although vaccine development and anti-inflammation therapy lower the severity of COVID-19, the prevalence and mortality rates are still alarming in some countries worldwide. In this review, we describe the involvement of viral proteins in activating ER stress transducers and their downstream signals and in inducing inflammation and inflammasome formation. Furthermore, we propose the potential of melatonin as an ER stress modulator, owing to its antioxidant, anti-inflammatory, and immunoregulatory effects in viral infections. Considering its strong safety profile, we suggest that additive melatonin supplementation in the elderly could be beneficial in treating COVID-19

Parenting stress in parents of children with refractory epilepsy before and after vagus nerve stimulation implantation

Objectives: The purpose of this study was to evaluate parenting stress in parents of children with refractory epilepsy before and after their children received vagus nerve stimulation (VNS) implantation. Methods: Parents of children with refractory epilepsy completed the Parenting Stress Index (PSI) under a psychologist's assessment before and at least 12 months after their children received VNS implantation. The PSI questionnaire measures parenting stress in two domains; a parent domain with seven subscales, and a child domain with six. Age, gender, epilepsy comorbidity, VNS implantation date, seizure frequency, and anticonvulsant history before and after VNS implantation were obtained from reviews of medical charts. Results: In total, 30 parents completed the first and follow-up PSI questionnaires. Seventeen of their children (56.7%) were boys. The children aged from 1 to 12 years (7.43 ± 3.59 years, mean ± SD). After VNS implantation, the mean total parenting stress scores decreased from 282.1 ± 38.0 to 272.4 ± 42.9. A significant decrease was found on the spouse subscale of the parent domain. For the parents of boys, the mean total parenting stress scores decreased significantly. The mean total parenting stress scores also decreased significantly for parents of epileptic children without autism and who did not taper off the number of different anticonvulsants used after VNS. Conclusions: VNS is an advisable choice to treat refractory epilepsy. Our study showed that 12 months or more after VNS implantation, seizure frequency and parenting stress typically decreased. However, in some special cases the parenting stress may increase, and external help may be required to support these patients and their parents. Key Words: children, refractory epilepsy, parenting stress, vagus nerve stimulatio