Given a graph G, denote by ∆, d and χ ′ the maximum degree, the average degree and the chromatic index of G,66, 0.69392∆ − 0.20642 if ∆ = 65, and 0.68706∆ + 0.19815 if 56 ≤ ∆ ≤ 64. This result improves the best known bound 2 3 (∆ + 2) obtained by Woodall in 2007 for ∆ ≥ 56. Additionally, Woodall constructed an infinite family of graphs showing his result cannot be improved by well-known Vizing's Adjacency Lemma and other known edge-coloring techniques. To over come the barrier, we follow the recently developed recoloring technique of Tashkinov trees to expand Vizing fans technique to a larger class of trees.

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Background: To probe into the influence of evidence-based nursing (EBN) on clinical recovery and prognosis of patients with arrhythmia after acute myocardial infarction (AMI). Methods: Totally, 240 AMI patients with arrhythmia treated in Taizhou People's Hospital（Jiangsu, China） from July 2019 to December 2020 were collected and randomly divided into the study group (n = 120) and control group (n = 120). The control group was received routine nursing, while the study group carried out EBN. The following indicators were evaluated and compared between the two groups: length of hospital stay, symptom disappearance time, cardiac function, psychological status, and incidence of adverse events after 6 months of follow-up were. Results: Compared to the control group, the length of hospital stay, symptom disappearance time, LVEF (left ventricular ejection fraction), LVEDD (left ventricular end-diastolic diameter), SAS (self-rating anxiety scale) score and SDS (self-rating depression scale) score in the study group were significant improves (P < 0.05), and the incidence of adverse events after 6-month follow-up in the study group was also significantly lower than that in the control group (P < 0.05). Conclusion: EBN intervention for AMI patients with arrhythmia can significantly improve the length of hospital stay and symptom disappearance time, adjust cardiac function and psychological status, and reduce the incidence of adverse events.

Given a graph G, denote by ∆ and χ ′ the maximum degree and the chromatic index of G, respectively. A simple graph G is called edge-∆-critical if χ ′ (G) = ∆ + 1 and χ ′ (H) ≤ ∆ for every proper subgraph H of G. We proved that every edge chromatic critical graph of order n with maximum degree at least 2n 3 +12 is Hamiltonian.