42,885 research outputs found
On some intriguing problems in Hamiltonian graph theory -- A survey
We survey results and open problems in Hamiltonian graph theory centred around three themes: regular graphs, -tough graphs, and claw-free graphs
Hamilton cycles in graphs and hypergraphs: an extremal perspective
As one of the most fundamental and well-known NP-complete problems, the
Hamilton cycle problem has been the subject of intensive research. Recent
developments in the area have highlighted the crucial role played by the
notions of expansion and quasi-randomness. These concepts and other recent
techniques have led to the solution of several long-standing problems in the
area. New aspects have also emerged, such as resilience, robustness and the
study of Hamilton cycles in hypergraphs. We survey these developments and
highlight open problems, with an emphasis on extremal and probabilistic
approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page
limits, this final version is slightly shorter than the previous arxiv
versio
Modularity bounds for clusters located by leading eigenvectors of the normalized modularity matrix
Nodal theorems for generalized modularity matrices ensure that the cluster
located by the positive entries of the leading eigenvector of various
modularity matrices induces a connected subgraph. In this paper we obtain lower
bounds for the modularity of that set of nodes showing that, under certain
conditions, the nodal domains induced by eigenvectors corresponding to highly
positive eigenvalues of the normalized modularity matrix have indeed positive
modularity, that is they can be recognized as modules inside the network.
Moreover we establish Cheeger-type inequalities for the cut-modularity of the
graph, providing a theoretical support to the common understanding that highly
positive eigenvalues of modularity matrices are related with the possibility of
subdividing a network into communities
Synchronous Context-Free Grammars and Optimal Linear Parsing Strategies
Synchronous Context-Free Grammars (SCFGs), also known as syntax-directed
translation schemata, are unlike context-free grammars in that they do not have
a binary normal form. In general, parsing with SCFGs takes space and time
polynomial in the length of the input strings, but with the degree of the
polynomial depending on the permutations of the SCFG rules. We consider linear
parsing strategies, which add one nonterminal at a time. We show that for a
given input permutation, the problems of finding the linear parsing strategy
with the minimum space and time complexity are both NP-hard
Connectivity and Cycles in Graphs
https://digitalcommons.memphis.edu/speccoll-faudreerj/1199/thumbnail.jp
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