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Uniform coverings of 2-paths with 5-paths in the complete graph

Midori Kobayashi, Gisaku Nakamura, Chie Nara
2005 Discrete Mathematics  
There exists a uniform covering of 2-paths with 5-paths in K n if and only if n is even or n ≡ 1 (mod 8).  ...  Acknowledgements The authors thank the referees for their valuable comments. References  ...  There exists a uniform covering of 2-paths with 5-paths in K n . Let k 1. From Propositions 2.1 and 2.2, there exist uniform coverings of 2-paths with 5-paths in K 8k and K 9 .  ... 
doi:10.1016/j.disc.2004.03.019 fatcat:lpus6fbi4rhsvem23qzg7vpzwi

Vertex covers by monochromatic pieces — A survey of results and problems

András Gyárfás
2016 Discrete Mathematics  
This survey is devoted to problems and results concerning covering the vertices of edge colored graphs or hypergraphs with monochromatic paths, cycles and other objects.  ...  It is an expanded version of the talk with the same title at the Seventh Cracow Conference on Graph Theory, held in Rytro in September 14-19, 2014.  ...  Thanks for Gábor Sárközy for his remarks on the manuscript. The author is also indebted to referees for valuable remarks.  ... 
doi:10.1016/j.disc.2015.07.007 fatcat:sssibblukngrxhnxjeeiut5rfy

Monochromatic Path and Cycle Partitions in Hypergraphs

András Gyárfás, Gábor N. Sárközy
2013 Electronic Journal of Combinatorics  
Also, apart from $2k-5$ vertices, $2$-colored $k$-uniform hypergraphs can be partitioned into two monochromatic loose paths.In general, we prove that in any $r$-coloring of a $k$-uniform hypergraph there  ...  Here we address the problem to partition edge colored hypergraphs by monochromatic paths and cycles generalizing a well-known similar problem for graphs.We show that $r$-colored $r$-uniform complete hypergraphs  ...  In any 2-coloring of the edges of a finite complete graph the vertices can be partitioned into a red and a blue path. Here the empty graph and the one-vertex graph is accepted as a path of any color.  ... 
doi:10.37236/2631 fatcat:hmm5ntcssrebrl5frgc5vugz54

Monochromatic loose path partitions in k-uniform hypergraphs [article]

Changhong Lu, Bing Wang, Ping Zhang
2016 arXiv   pre-print
A conjecture of Gyárfás and Sárközy says that in every 2-coloring of the edges of the complete k-uniform hypergraph K_n^k, there are two disjoint monochromatic loose paths of distinct colors such that  ...  A weaker form of this conjecture with 2k-5 uncovered vertices instead of k-2 is proved, thus the conjecture holds for k=3.  ...  Proposition 1.1 In any 2-coloring of the edges of a finite complete graph the vertices can be partitioned into a red and a blue path.  ... 
arXiv:1611.03259v1 fatcat:vmqxhlickre5rgalzrwy5r5uw4

Monochromatic paths in 2-edge coloured graphs and hypergraphs [article]

Maya Stein
2022 arXiv   pre-print
Further, we give a lower bound for the number of tight paths needed to partition any 2-edge-coloured complete k-partite k-uniform hypergraph, and show that any 2-edge coloured bipartite graph has a partition  ...  We answer a question of Gyárfás and Sárközy from 2013 by showing that every 2-edge-coloured complete 3-uniform hypergraph can be partitioned into two monochromatic tight paths of different colours.  ...  A variant of the original problem in graphs replaces the underlying complete graph with the complete balanced bipartite graph K n,n .  ... 
arXiv:2204.12464v1 fatcat:wyiy5koju5e45ndvm4qmanl34u

On Hamiltonian Berge cycles in 3-uniform hypergraphs [article]

Linyuan Lu, Zhiyu Wang
2020 arXiv   pre-print
Given a set R, a hypergraph is R-uniform if the size of every hyperedge belongs to R. A hypergraph H is called covering if every vertex pair is contained in some hyperedge in H.  ...  As an application, we determine the maximum Lagrangian of k-uniform Berge-C_t-free hypergraphs and Berge-P_t-free hypergraphs.  ...  This completes the proof of Theorem 7. Proof of Corollary 9 and 10. For k = 3, Corollary 9 and 10 follows from Theorem 4 and Lemma 19 using the same logic as the proof of Theorem 7.  ... 
arXiv:1901.06042v3 fatcat:l6g7xzsluzdsdkd4ohub32ojwe

Page 8951 of Mathematical Reviews Vol. , Issue 2003m [page]

2003 Mathematical Reviews  
-EDD; Shibuya) Uniform coverings of 2-paths with 5-paths in K>,,.  ...  Summary: “We give a construction of a uniform covering of 2- paths with 5-paths in K,, for all even n > 6, i.e., we construct a set S of 5-paths in K,, having the property that each 2-path in K,, lies  ... 

Labeling, Covering And Decomposing Of Graphs — Smarandache'S Notion In Graph Theory

Linfan Mao
2010 Zenodo  
This paper surveys the applications of Smarandache's notion to graph theory appeared in International J.Math.Combin. from Vol.1,2008 to Vol.3,2009.  ...  In fact, many problems discussed in these papers are generalized in this paper.  ...  my monograph Graph Theory -A Smarandachely Type will be appeared in 2012.  ... 
doi:10.5281/zenodo.9258 fatcat:rqsysf4dqreslozhxbiljqfbrm

Labeling, Covering And Decomposing Of Graphs — Smarandache'S Notion In Graph Theory

Linfan Mao
2010 Zenodo  
This paper surveys the applications of Smarandache's notion to graph theory appeared in International J.Math.Combin. from Vol.1,2008 to Vol.3,2009.  ...  In fact, many problems discussed in these papers are generalized in this paper.  ...  my monograph Graph Theory -A Smarandachely Type will be appeared in 2012.  ... 
doi:10.5281/zenodo.823874 fatcat:mzzws7nuijb33norbr5ifpzgh4

Page 3572 of Mathematical Reviews Vol. , Issue 83i [page]

1983 Mathematical Reviews  
The graph is then of type (r-K,.--+2),3 in other words, there are rc—c+2 classes with r vertices, the graph induced by two classes is a matching, and between two vertices at distance 2, there are c paths  ...  Jung (Berlin) Kohler, Norbert 83i:05052 On path-coverings and Hamilton-connectivity of finite graphs. Arch. Math. ( Basel) 36 (1981), no. 5, 470-473.  ... 

Searching symmetric networks with Utilitarian-Postman paths

Steve Alpern, Vic Baston, Shmuel Gal
2009 Networks  
We then show that if Q is symmetric of odd degree, with n vertices and m unit length edges, the value V of (Q) satis...es V m 2 + n 2 2n 8m ; with equality if and only if Q has a path v 1 ; v 2 ; : : :  ...  For any network Q; one may consider the zero-sum search game (Q) in which the (minimizing) Searcher picks a unit speed path S (t) in Q; the Hider picks a point H in Q; and the payo¤ is the meeting time  ...  In particular, all complete graphs, complete bipartite graphs, hypercube graphs, graphs of large valency, are HHHC. And yes, the Petersen graph is HHHC.  ... 
doi:10.1002/net.20314 fatcat:m4hvsrkfgnhoxmvkg6pde3vpca

Almost partitioning 2-coloured complete 3-uniform hypergraphs into two monochromatic tight or loose cycles [article]

Sebastian Bustamante, Hiep Han, Maya Stein
2018 arXiv   pre-print
We show that for every η > 0 there exists an integer n_0 such that every 2-colouring of the 3-uniform complete hypergraph on n ≥ n_0 vertices contains two disjoint monochromatic tight cycles of distinct  ...  The same result holds if we replace tight cycles with loose cycles.  ...  It states that for every n, every 2-colouring of the edges of the complete graph K n admits a covering of the vertices of K n with at most two monochromatic vertex disjoint cycles of different colours.  ... 
arXiv:1701.07806v2 fatcat:wcqrmb6uyrcmtju53a6bztgxxu

Finding paths with minimum shared edges

Masoud T. Omran, Jörg-Rüdiger Sack, Hamid Zarrabi-Zadeh
2012 Journal of combinatorial optimization  
to t in G so as to minimize the number of edges shared among the paths.  ...  Motivated by a security problem in geographic information systems, we study the following graph theoretical problem: given a graph G, two special nodes s and t in G, and a number k, find k paths from s  ...  We used High Performance Computing Virtual Laboratory (HPCVL)'s Beowulf Cluster 3 that has 64 nodes of 4×2.2 GHz Opteron Cores with 8 GB RAM for running the experiments in parallel.  ... 
doi:10.1007/s10878-012-9462-2 fatcat:7szvmvpqkrh5ffgpmysivmqswi

Finding Paths with Minimum Shared Edges [chapter]

Masoud T. Omran, Jörg-Rüdiger Sack, Hamid Zarrabi-Zadeh
2011 Lecture Notes in Computer Science  
to t in G so as to minimize the number of edges shared among the paths.  ...  Motivated by a security problem in geographic information systems, we study the following graph theoretical problem: given a graph G, two special nodes s and t in G, and a number k, find k paths from s  ...  We used High Performance Computing Virtual Laboratory (HPCVL)'s Beowulf Cluster 3 that has 64 nodes of 4×2.2 GHz Opteron Cores with 8 GB RAM for running the experiments in parallel.  ... 
doi:10.1007/978-3-642-22685-4_49 fatcat:4h5fktr2cbba3clvfqx57mebym

Partial barrier coverage: Using game theory to optimize probability of undetected intrusion in polygonal environments

Stephen Kloder, Seth Hutchinson
2008 2008 IEEE International Conference on Robotics and Automation  
We use ideas from noncooperative game theory together with previous results from complete barrier coverage -the problem of completely preventing undetected intrusion -to develop new methods that solve  ...  In this paper, we formalize the problem of partial barrier coverage, that is, the problem of using robot sensors (guards) to minimize the probability of undetected intrusion in a particular region by an  ...  A complete barrier covers all intrusion paths. A partial barrier covers an infinite set of paths, and misses an infinite set of paths.  ... 
doi:10.1109/robot.2008.4543615 dblp:conf/icra/KloderH08 fatcat:6ya5vg2yyvdmdotr64npqcswjq
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