|
| ||
|
|
||
Bipartite Graph G (bipartite + graph_g)
Selected AbstractsIsomorphism criterion for monomial graphsJOURNAL OF GRAPH THEORY, Issue 4 2005Vasyl Dmytrenko Abstract Let q be a prime power, ,,q be the field of q elements, and k,,m be positive integers. A bipartite graph G,=,Gq(k,,m) is defined as follows. The vertex set of G is a union of two copies P and L of two-dimensional vector spaces over ,,q, with two vertices (p1,,p2) , P and [ l1,,l2] , L being adjacent if and only if p2,+,l2,=,pl. We prove that graphs Gq(k,,m) and Gq,(k,,,m,) are isomorphic if and only if q,=,q, and {gcd,(k,,q,,,1), gcd,(m,,q,,,1)},=,{gcd,(k,,,q,,,1),gcd,(m,,,q,,,1)} as multisets. The proof is based on counting the number of complete bipartite INFgraphs in the graphs. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 322,328, 2005 [source] Mutually independent hamiltonian paths in star networksNETWORKS: AN INTERNATIONAL JOURNAL, Issue 2 2005Cheng-Kuan Lin Abstract Two hamiltonian paths P1 = ,u1, u2,,,un(G), and P2 = ,v1, v2,,,vn(G), of G from u to v are independent if u = u1 = v1, v = vn(G) = un(G), and vi , ui for every 1 < i < n(G). A set of hamiltonian paths, {P1, P2,,,Pk}, of G from u to v are mutually independent if any two different hamiltonian paths are independent from u to v. A bipartite graph G is hamiltonian laceable if there exists a hamiltonian path joining any two nodes from different partite sets. A bipartite graph is k -mutually independent hamiltonian laceable if there exists k -mutually independent hamiltonian paths between any two nodes from distinct partite sets. The mutually independent hamiltonian laceability of a bipartite graph G, IHPL(G), is the maximum integer k such that G is k -mutually independent hamiltonian laceable. Let Sn denote the n -dimensional star graph. We prove that IHPL(S2) = 1, IHPL(S3) = 0, and IHPL(Sn) = n, 2 if n , 4. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 46(2), 110,117 2005 [source] Space complexity of random formulae in resolutionRANDOM STRUCTURES AND ALGORITHMS, Issue 1 2003Eli Ben-Sasson We study the space complexity of refuting unsatisfiable random k -CNFs in the Resolution proof system. We prove that for , , 1 and any , > 0, with high probability a random k -CNF over n variables and ,n clauses requires resolution clause space of ,(n/,1+,). For constant ,, this gives us linear, optimal, lower bounds on the clause space. One consequence of this lower bound is the first lower bound for size of treelike resolution refutations of random 3-CNFs with clause density , , n. This bound is nearly tight. Specifically, we show that with high probability, a random 3-CNF with ,n clauses requires treelike refutation size of exp(,(n/,1+,)), for any , > 0. Our space lower bound is the consequence of three main contributions: (1) We introduce a 2-player Matching Game on bipartite graphs G to prove that there are no perfect matchings in G. (2) We reduce lower bounds for the clause space of a formula F in Resolution to lower bounds for the complexity of the game played on the bipartite graph G(F) associated with F. (3) We prove that the complexity of the game is large whenever G is an expander graph. Finally, a simple probabilistic analysis shows that for a random formula F, with high probability G(F) is an expander. We also extend our result to the case of G-PHP, a generalization of the Pigeonhole principle based on bipartite graphs G. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 92,109, 2003 [source] | ||||||||||