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## Ordered Pairs (ordered + pair)
## Selected Abstracts## Extremal solutions for nonlinear second order differential inclusions MATHEMATISCHE NACHRICHTEN, Issue 1-2 2005P. DoukaAbstract We consider a nonlinear second order differential inclusion driven by the scalar p -Laplacian and with nonlinear multivalued boundary conditions. Assuming the existence of an ordered pair of upper-lower solutions and using truncation and penalization techniques together with Zorn's lemma, we show that the problem has extremal solutions in the order interval formed by the upper und lower solutions. We present some special cases of interest and show that our method applies also to the periodic problem. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## A phase transition for avoiding a giant component RANDOM STRUCTURES AND ALGORITHMS, Issue 2 2006Tom BohmanLet c be a constant and (e1,f1),(e2,f2),,,(ecn,fcn) be a sequence of ordered pairs of edges from the complete graph Kn chosen uniformly and independently at random. We prove that there exists a constant c2 such that if c > c2, then whp every graph which contains at least one edge from each ordered pair (ei,fi) has a component of size ,(n), and, if c < c2, then whp there is a graph containing at least one edge from each pair that has no component with more than O(n1,, vertices, where , is a constant that depends on c2 , c. The constant c2 is roughly 0.97677. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006 [source] ## The proof of a conjecture of Bouabdallah and Sotteau NETWORKS: AN INTERNATIONAL JOURNAL, Issue 4 2004Min XuAbstract Let G be a connected graph of order n. A routing in G is a set of n(n , 1) fixed paths for all ordered pairs of vertices of G. The edge-forwarding index of G, ,(G), is the minimum of the maximum number of paths specified by a routing passing through any edge of G taken over all routings in G, and ,,,n is the minimum of ,(G) taken over all graphs of order n with maximum degree at most ,. To determine ,n,2p,1,n for 4p + 2,p/3, + 1 , n , 6p, A. Bouabdallah and D. Sotteau proposed the following conjecture in [On the edge forwarding index problem for small graphs, Networks 23 (1993), 249,255]. The set 3 × {1, 2, , , ,(4p)/3,} can be partitioned into 2p pairs plus singletons such that the set of differences of the pairs is the set 2 × {1, 2, , , p}. This article gives a proof of this conjecture and determines that ,n,2p,1,n is equal to 5 if 4p + 2,p/3, + 1 , n , 6p and to 8 if 3p + ,p/3, + 1 , n , 3p + ,(3p)/5, for any p , 2. © 2004 Wiley Periodicals, Inc. NETWORKS, Vol. 44(4), 292,296 2004 [source] ## A phase transition for avoiding a giant component RANDOM STRUCTURES AND ALGORITHMS, Issue 2 2006Tom BohmanLet c be a constant and (e1,f1),(e2,f2),,,(ecn,fcn) be a sequence of ordered pairs of edges from the complete graph Kn chosen uniformly and independently at random. We prove that there exists a constant c2 such that if c > c2, then whp every graph which contains at least one edge from each ordered pair (ei,fi) has a component of size ,(n), and, if c < c2, then whp there is a graph containing at least one edge from each pair that has no component with more than O(n1,, vertices, where , is a constant that depends on c2 , c. The constant c2 is roughly 0.97677. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006 [source] ## Avoiding a giant component RANDOM STRUCTURES AND ALGORITHMS, Issue 1 2001Tom BohmanLet e1,,e,1; e2,,e,2;,;ei,,e,i;,,, be a sequence of ordered pairs of edges chosen uniformly at random from the edge set of the complete graph Kn (i.e. we sample with replacement). This sequence is used to form a graph by choosing at stage i, i=1,,, one edge from ei,e,i to be an edge in the graph, where the choice at stage i is based only on the observation of the edges that have appeared by stage i. We show that these choices can be made so that whp the size of the largest component of the graph formed at stage 0.535n is polylogarithmic in n. This resolves a question of Achlioptas. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19, 75,85, 2001 [source] |