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

An extremal problem on group connectivity of graphs

Let A be an Abelian group, n \u3e 3 be an integer, and ex(n, A) be the maximum integer such that every n-vertex simple graph with at most ex(n, A) edges is not A-connected. In this paper, we study ex(n, A) for IAI \u3e 3 and present lower and upper bounds for 3 \u3c IAI 5. 0 2012 Elsevier Ltd. All rights reserved

Hamilton-chain saturated hypergraphs

AbstractWe say that a hypergraph H is hamiltonian path (cycle) saturated if H does not contain an open (closed) hamiltonian chain but by adding any new edge we create an open (closed) hamiltonian chain in H. In this paper we ask about the smallest size of an r-uniform hamiltonian path (cycle) saturated hypergraph, mainly for r=3. We present a construction of a family of 3-uniform path (cycle) saturated hamiltonian hypergraphs with O(n5/2) edges. On the other hand we prove that the number of edges in an r-uniform hamiltonian path (cycle) saturated hypergraph is at least Ω(nr−1)

Group Connectivity of Graphs

Tutte introduced the theory of nowhere-zero flows and showed that a plane graph G has a face k-coloring if and only if G has a nowhere-zero A-flow, for any Abelian group A with |A| ≥ k. In 1992 Jaeger et al. [16] extended nowhere-zero flows to group connectivity of graphs: given an orientation D of a graph G, if for any b: V (G) A with sumv∈ V(G ) b(v) = 0, there always exists a map ƒ: E(G) A - {lcub}0{rcub}, such that at each v ∈ V(G), e=vw isdirectedfrom vtow fe- e=uvi sdirectedfrom utov fe=b v in A, then G is A-connected. For a 2-edge-connected graph G, define Lambda g(G) = min{lcub}k: for any Abelian group A with |A| ≥ k, G is A-connected{rcub}.;Let G1 ⊗ G2 and G1 xG2 denote the strong and Cartesian product of two connected nontrivial graphs G1 and G2. We prove that Lambdag(G 1 ⊗ G2) ≤ 4, where equality holds if and only if both G1 and G 2 are trees and min{lcub}|V (G1)|, |V (G2)|{rcub}=2; Lambda g(G1 ⊗ G 2) ≤ 5, where equality holds if and only if both G 1 and G2 are trees and either G 1 ≅ K1, m and G2 ≅ K 1,n, for n, m ≥ 2 or min{lcub}|V (G1)|, | V (G2)|{rcub}=2. A similar result for the lexicographical product graphs is also obtained.;Let P denote a path in G, let beta G(P) be the minimum length of a circuit containing P, and let betai(G) be the maximum of betaG(P) over paths of length i in G. We show that Lambda g(G) ≤ betai( G) + 1 for any integer i \u3e 0 and for any 2-connected graph G. Partial solutions toward determining the graphs for which equality holds were obtained by Fan et al. in [J. Comb. Theory, Ser. B, 98(6) (2008), 1325-1336], among others. We completely determine all graphs G with Lambda g(G) = beta2(G) + 1.;Let Z3 denote the cyclic group of order 3. In [16], Jaeger et al. conjectured that every 5-edge-connected graph is Z3 -connected. We proved the following: (i) Every 5-edge-connected graph is Z3 -connected if and only if every 5-edge-connected line graph is Z3 -connected. (ii) Every 6-edge-connected triangular line graph is Z3 -connected. (iii) Every 7-edge-connected triangular claw-free graph is Z3 -connected. In particular, every 6-edge-connected triangular line graph and every 7-edge-connected triangular claw-free graph have a nowhere-zero 3-flow