A Victorian Age Proof of the Four Color Theorem

In this paper we have investigated some old issues concerning four color map problem. We have given a general method for constructing counter-examples to Kempe's proof of the four color theorem and then show that all counterexamples can be rule out by re-constructing special 2-colored two paths decomposition in the form of a double-spiral chain of the maximal planar graph. In the second part of the paper we have given an algorithmic proof of the four color theorem which is based only on the coloring faces (regions) of a cubic planar maps. Our algorithmic proof has been given in three steps. The first two steps are the maximal mono-chromatic and then maximal dichromatic coloring of the faces in such a way that the resulting uncolored (white) regions of the incomplete two-colored map induce no odd-cycles so that in the (final) third step four coloring of the map has been obtained almost trivially.Comment: 27 pages, 18 figures, revised versio

On a combination of the 1-2-3 conjecture and the antimagic labelling conjecture

International audienceThis paper is dedicated to studying the following question: Is it always possible to injectively assign the weights 1, ..., |E(G)| to the edges of any given graph G (with no component isomorphic to K2) so that every two adjacent vertices of G get distinguished by their sums of incident weights? One may see this question as a combination of the well-known 1-2-3 Conjecture and the Antimagic Labelling Conjecture. Throughout this paper, we exhibit evidence that this question might be true. Benefiting from the investigations on the Antimagic Labelling Conjecture, we first point out that several classes of graphs, such as regular graphs, indeed admit such assignments. We then show that trees also do, answering a recent conjecture of Arumugam, Premalatha, Bača and Semaničová-Feňovčíková. Towards a general answer to the question above, we then prove that claimed assignments can be constructed for any graph, provided we are allowed to use some number of additional edge weights. For some classes of sparse graphs, namely 2-degenerate graphs and graphs with maximum average degree 3, we show that only a small (constant) number of such additional weights suffices

European Journal of Combinatorics Index, Volume 27

BACKGROUND: Diabetes is an inflammatory condition associated with iron abnormalities and increased oxidative damage. We aimed to investigate how diabetes affects the interrelationships between these pathogenic mechanisms. METHODS: Glycaemic control, serum iron, proteins involved in iron homeostasis, global antioxidant capacity and levels of antioxidants and peroxidation products were measured in 39 type 1 and 67 type 2 diabetic patients and 100 control subjects. RESULTS: Although serum iron was lower in diabetes, serum ferritin was elevated in type 2 diabetes (p = 0.02). This increase was not related to inflammation (C-reactive protein) but inversely correlated with soluble transferrin receptors (r = - 0.38, p = 0.002). Haptoglobin was higher in both type 1 and type 2 diabetes (p &lt; 0.001) and haemopexin was higher in type 2 diabetes (p &lt; 0.001). The relation between C-reactive protein and haemopexin was lost in type 2 diabetes (r = 0.15, p = 0.27 vs r = 0.63, p &lt; 0.001 in type 1 diabetes and r = 0.36, p = 0.001 in controls). Haemopexin levels were independently determined by triacylglycerol (R(2) = 0.43) and the diabetic state (R(2) = 0.13). Regarding oxidative stress status, lower antioxidant concentrations were found for retinol and uric acid in type 1 diabetes, alpha-tocopherol and ascorbate in type 2 diabetes and protein thiols in both types. These decreases were partially explained by metabolic-, inflammatory- and iron alterations. An additional independent effect of the diabetic state on the oxidative stress status could be identified (R(2) = 0.5-0.14). CONCLUSIONS: Circulating proteins, body iron stores, inflammation, oxidative stress and their interrelationships are abnormal in patients with diabetes and differ between type 1 and type 2 diabetes</p