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91.
In the game of cops and robber, the cops try to capture a robber moving on the vertices of the graph. The minimum number of cops required to win on a given graph G is called the cop number of G. The biggest open conjecture in this area is the one of Meyniel, which asserts that for some absolute constant C, the cop number of every connected graph G is at most . In a separate paper, we showed that Meyniel's conjecture holds asymptotically almost surely for the binomial random graph. The result was obtained by showing that the conjecture holds for a general class of graphs with some specific expansion‐type properties. In this paper, this deterministic result is used to show that the conjecture holds asymptotically almost surely for random d‐regular graphs when d = d(n) ≥ 3. 相似文献
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Eckhard Steffen 《Journal of Graph Theory》2015,78(3):195-206
Let G be a bridgeless cubic graph. Consider a list of k 1‐factors of G. Let be the set of edges contained in precisely i members of the k 1‐factors. Let be the smallest over all lists of k 1‐factors of G. Any list of three 1‐factors induces a core of a cubic graph. We use results on the structure of cores to prove sufficient conditions for Berge‐covers and for the existence of three 1‐factors with empty intersection. Furthermore, if , then is an upper bound for the girth of G. We also prove some new upper bounds for the length of shortest cycle covers of bridgeless cubic graphs. Cubic graphs with have a 4‐cycle cover of length and a 5‐cycle double cover. These graphs also satisfy two conjectures of Zhang 18 . We also give a negative answer to a problem stated in 18 . 相似文献
96.
We prove part of a conjecture by Johansson, Kahn, and Vu (Factors in random graphs, Random Struct. Algorithms 33 (2008), 1, 1–28.) regarding threshold functions for the existence of an H‐factor in a random graph . We prove that the conjectured threshold function is correct for any graph H which is not covered by its densest subgraphs. We also demonstrate that the main result of Johansson, Kahn, and Vu (Factors in random graphs, Random Struct. Algorithms 33 (2008), 1, 1–28) generalizes to multigraphs, digraphs, and a multipartite model. 相似文献
97.
The independence number of a sparse random graph G(n,m) of average degree d = 2m/n is well‐known to be with high probability, with in the limit of large d. Moreover, a trivial greedy algorithm w.h.p. finds an independent set of size , i.e., about half the maximum size. Yet in spite of 30 years of extensive research no efficient algorithm has emerged to produce an independent set with size for any fixed (independent of both d and n). In this paper we prove that the combinatorial structure of the independent set problem in random graphs undergoes a phase transition as the size k of the independent sets passes the point . Roughly speaking, we prove that independent sets of size form an intricately rugged landscape, in which local search algorithms seem to get stuck. We illustrate this phenomenon by providing an exponential lower bound for the Metropolis process, a Markov chain for sampling independent sets. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 436–486, 2015 相似文献
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Hanna Talacua Serge H. M. Sntjens Shraddha H. Thakkar Aurelie M. A. Brizard Lex A. van Herwerden Aryan Vink Geert C. van Almen Patricia Y. W. Dankers Carlijn V. C. Bouten Ricardo P. J. Budde Henk M. Janssen Jolanda Kluin 《Macromolecular bioscience》2020,20(7)
For in situ tissue engineering (TE) applications it is important that implant degradation proceeds in concord with neo‐tissue formation to avoid graft failure. It will therefore be valuable to have an imaging contrast agent (CA) available that can report on the degrading implant. For this purpose, a biodegradable radiopaque biomaterial is presented, modularly composed of a bisurea chain‐extended polycaprolactone (PCL2000‐U4U) elastomer and a novel iodinated bisurea‐modified CA additive (I‐U4U). Supramolecular hydrogen bonding interactions between the components ensure their intimate mixing. Porous implant TE‐grafts are prepared by simply electrospinning a solution containing PCL2000‐U4U and I‐U4U. Rats receive an aortic interposition graft, either composed of only PCL2000‐U4U (control) or of PCL2000‐U4U and I‐U4U (test). The grafts are explanted for analysis at three time points over a 1‐month period. Computed tomography imaging of the test group implants prior to explantation shows a decrease in iodide volume and density over time. Explant analysis also indicates scaffold degradation. (Immuno)histochemistry shows comparable cellular contents and a similar neo‐tissue formation process for test and control group, demonstrating that the CA does not have apparent adverse effects. A supramolecular approach to create solid radiopaque biomaterials can therefore be used to noninvasively monitor the biodegradation of synthetic implants. 相似文献
100.
We give a characterization of the cyclic subgroup separability and weak potency of the fundamental group of a graph of polycyclic-by-finite groups and free-by-finite groups amalgamating edge subgroups of the form × D,where h has infinite order and D is finite. 相似文献