Linked graphs with restricted lengths |
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Authors: | Guantao Chen Yuan Chen Shuhong Gao Zhiquan Hu |
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Institution: | aDepartment of Mathematics and Statistics, Georgia State University, Atlanta, GA 30303, USA;bFaculty of Mathematics and Computer Science, Hubei University, Wuhan, China;cCollege of Science, Wuhan University of Science and Engineering, Wuhan, China;dDepartment of Mathematical Sciences, Clemson University, Clemson, SC 29634, USA;eFaculty of Mathematics and Statistics, Central China Normal University, Wuhan, China |
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Abstract: | A graph G is k-linked if G has at least 2k vertices, and for every sequence x1,x2,…,xk,y1,y2,…,yk of distinct vertices, G contains k vertex-disjoint paths P1,P2,…,Pk such that Pi joins xi and yi for i=1,2,…,k. Moreover, the above defined k-linked graph G is modulo (m1,m2,…,mk)-linked if, in addition, for any k-tuple (d1,d2,…,dk) of natural numbers, the paths P1,P2,…,Pk can be chosen such that Pi has length di modulo mi for i=1,2,…,k. Thomassen showed that, for each k-tuple (m1,m2,…,mk) of odd positive integers, there exists a natural number f(m1,m2,…,mk) such that every f(m1,m2,…,mk)-connected graph is modulo (m1,m2,…,mk)-linked. For m1=m2=…=mk=2, he showed in another article that there exists a natural number g(2,k) such that every g(2,k)-connected graph G is modulo (2,2,…,2)-linked or there is X V(G) such that |X| 4k−3 and G−X is a bipartite graph, where (2,2,…,2) is a k-tuple.We showed that f(m1,m2,…,mk) max{14(m1+m2+ +mk)−4k,6(m1+m2+ +mk)−4k+36} for every k-tuple of odd positive integers. We then extend the result to allow some mi be even integers. Let (m1,m2,…,mk) be a k-tuple of natural numbers and ℓ k such that mi is odd for each i with ℓ+1 i k. If G is 45(m1+m2+ +mk)-connected, then either G has a vertex set X of order at most 2k+2ℓ−3+δ(m1,…,mℓ) such that G−X is bipartite or G is modulo (2m1,…,2mℓ,mℓ+1,…,mk)-linked, where Our results generalize several known results on parity-linked graphs. |
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Keywords: | Connectivity Linkage Bipartite index Sumset |
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