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  • Author: Anna Ivanova x
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We prove that every planar graph with maximum degree ∆ is strong edge (2∆−1)-colorable if its girth is at least 40+1. The bound 2∆−1 is reached at any graph that has two adjacent vertices of degree ∆.


The weight w(f) of a face f in a 3-polytope is the degree-sum of vertices incident with f. It follows from Lebesgue’s results of 1940 that every triangle-free 3-polytope without 4-faces incident with at least three 3-vertices has a 4-face with w ≤ 21 or a 5-face with w ≤ 17. Here, the bound 17 is sharp, but it was still unknown whether 21 is sharp.

The purpose of this paper is to improve this 21 to 20, which is best possible.


proved that every 3-polytope P 5 of girth 5 has a path of three vertices of degree 3. refined this by showing that every P 5 has a 3-vertex with two 3-neighbors and the third neighbor of degree at most 4. This description of 3-stars in P 5s is tight in the sense that no its parameter can be strengthened due to the dodecahedron combined with the existence of a P 5 in which every 3-vertex has a 4-neighbor.

We give another tight description of 3-stars in P 5s: there is a vertex of degree at most 4 having three 3-neighbors. Furthermore, we show that there are only these two tight descriptions of 3-stars in P 5s.

Also, we give a tight description of stars with at least three rays in P 5s and pose a problem of describing all such descriptions. Finally, we prove a structural theorem about P 5s that might be useful in further research.


Wilson’s disease is an inherited autosomal recessive disorder of copper balance leading to accumulation of copper mainly in liver and brain result from absent or reduced function of copper-transporting P-type ATPase. Copper is an essential trace element but in Wilson’s disease it accumulate to the point of toxicity. D-penicillamine is a classic drug for treatment of Wilson’s disease. Its major effect is to promote the urinary copper excretion. The use of D-penicillamine in the therapy of Wilson’s disease is known to be complicated by the development of various glomerular diseases. In this report we describe the development of nephrotic syndrome after 2 years treatment with D-penicillamine in a 31-year-old male undergoing treatment for Wilson’s disease, with a prompt regression at the discontinuation of the drug. We present this case to draw attention to the rare complication as nephrotic syndrome in patients with Wilson’s disease under D-penicillamine treatment and possible underlying causes. It is strongly necessary the therapy and clinical condition of patients with Wilson’s disease to be monitoring regularly - we recommended monthly.


Radial forearm flap is a gold standard for oral soft tissue defect reconstruction after tumour ablative surgery of oral cancer in advanced stages. The main disadvantage of this flap is donor site morbidity. The goal of our study was to show versatility of lateral arm flap in 34 cases with different oral defects that were reconstructed after tumour ablation, and to analyse complications and donor site morbidity. Thirty-four patients with advanced stage oral cancer (T3 and T4) underwent tumour ablation with or without suspicious lymph node removal and with immediate reconstruction of oral defect with lateral arm flap. Analysis of complications and donor sites morbidity was carried out. The Michigan Hand Outcome Questionnaire was used to evaluate functional and esthetical donor site outcome during at least one year follow up. Thirty-one patients had successful free flap surgery with uneventful post-surgery period. Flap loss due to vascularity problems was in one case (2.9%). The flap success rate was 97.1%. The donor site was closed primarily in all cases and healed uneventfully. The Michigan Hand Outcome Score was average 94.30%. The lateral arm is an excellent choice for oral reconstruction after ablative tumour surgery. It is versatile, safe and reliable for oral reconstruction with very good functional and aesthetical donor site outcome.


It is known that there are normal plane maps M5 with minimum degree 5 such that the minimum degree-sum w(S5) of 5-stars at 5-vertices is arbitrarily large. In 1940, Lebesgue showed that if an M5 has no 4-stars of cyclic type (5, 6, 6, 5) centered at 5-vertices, then w(S5) ≤ 68. We improve this bound of 68 to 55 and give a construction of a (5, 6, 6, 5)-free M5 with w(S5) = 48


In 1940, Lebesgue gave an approximate description of the neighborhoods of 5-vertices in the class P5 of 3-polytopes with minimum degree 5. Given a 3-polytope P, by h5(P) we denote the minimum of the maximum degrees (height) of the neighborhoods of 5-vertices (minor 5-stars) in P. Recently, Borodin, Ivanova and Jensen showed that if a polytope P in P5 is allowed to have a 5-vertex adjacent to two 5-vertices and two more vertices of degree at most 6, called a (5, 5, 6, 6,∞)-vertex, then h5(P) can be arbitrarily large. Therefore, we consider the subclass P_ 5 of 3-polytopes in P5 that avoid (5, 5, 6, 6,∞)-vertices. For each P* in P*5 without vertices of degree from 7 to 9, it follows from Lebesgue’s Theorem that h5(P*) ≤ 17. Recently, this bound was lowered by Borodin, Ivanova, and Kazak to the sharp bound h5(P*) ≤ 15 assuming the absence of vertices of degree from 7 to 11 in P*. In this note, we extend the bound h5(P*) ≤ 15 to all P*s without vertices of degree from 7 to 9.


In 1940, Lebesgue proved that every 3-polytope contains a 5-vertex for which the set of degrees of its neighbors is majorized by one of the following sequences:

  • (6, 6, 7, 7, 7), (6, 6, 6, 7, 9), (6, 6, 6, 6, 11), (5, 6, 7, 7, 8), (5, 6, 6, 7, 12), (5, 6, 6, 8, 10), (5, 6, 6, 6, 17), (5, 5, 7, 7, 13), (5, 5, 7, 8, 10), (5, 5, 6, 7, 27), (5, 5, 6, 6, ∞), (5, 5, 6, 8, 15), (5, 5, 6, 9, 11), (5, 5, 5, 7, 41), (5, 5, 5, 8, 23), (5, 5, 5, 9, 17), (5, 5, 5, 10, 14), (5, 5, 5, 11, 13).

In this paper we prove that every 3-polytope without vertices of degree from 7 to 11 contains a 5-vertex for which the set of degrees of its neighbors is majorized by one of the following sequences: (5, 5, 6, 6, ∞), (5, 6, 6, 6, 15), (6, 6, 6, 6, 6), where all parameters are tight.


The height of a face in a 3-polytope is the maximum degree of its incident vertices, and the height of a 3-polytope, h, is the minimum height of its faces. A face is pyramidal if it is either a 4-face incident with three 3-vertices, or a 3-face incident with two vertices of degree at most 4. If pyramidal faces are allowed, then h can be arbitrarily large, so we assume the absence of pyramidal faces in what follows.

In 1940, Lebesgue proved that every quadrangulated 3-polytope has h ≤ 11. In 1995, this bound was lowered by Avgustinovich and Borodin to 10. Recently, Borodin and Ivanova improved it to the sharp bound 8.

For plane triangulation without 4-vertices, , confirming the Kotzig conjecture of 1979, proved that h ≤ 20, which bound is sharp. Later, proved that h ≤ 20 for all triangulated 3-polytopes. In 1996, Horňák and Jendrol’ proved for arbitrarily polytopes that h ≤ 23. Recently, Borodin and Ivanova obtained the sharp bounds 10 for trianglefree polytopes and 20 for arbitrary polytopes.

In this paper we prove that any polytope has a face of degree at most 10 with height at most 20, where 10 and 20 are sharp.


Background: Bronchial asthma is a heterogeneous disease that includes various subtypes. They may share similar clinical characteristics, but probably have different pathological mechanisms.

Aim: To identify phenotypes using cluster analysis in moderate to severe bronchial asthma and to compare differences in clinical, physiological, immunological and inflammatory data between the clusters.

Patients and methods: Forty adult patients with moderate to severe bronchial asthma out of exacerbation were included. All underwent clinical assessment, anthropometric measurements, skin prick testing, standard spirometry and measurement fraction of exhaled nitric oxide. Blood eosinophilic count, serum total IgE and periostin levels were determined. Two-step cluster approach, hierarchical clustering method and k-mean analysis were used for identification of the clusters.

Results: We have identified four clusters. Cluster 1 (n=14) - late-onset, non-atopic asthma with impaired lung function, Cluster 2 (n=13) - late-onset, atopic asthma, Cluster 3 (n=6) - late-onset, aspirin sensitivity, eosinophilic asthma, and Cluster 4 (n=7) - early-onset, atopic asthma.

Conclusions: Our study is the first in Bulgaria in which cluster analysis is applied to asthmatic patients. We identified four clusters. The variables with greatest force for differentiation in our study were: age of asthma onset, duration of diseases, atopy, smoking, blood eosinophils, nonsteroidal anti-inflammatory drugs hypersensitivity, baseline FEV1/FVC and symptoms severity. Our results support the concept of heterogeneity of bronchial asthma and demonstrate that cluster analysis can be an useful tool for phenotyping of disease and personalized approach to the treatment of patients.