Highly parity linked graphs

Abstract

A graph G is k-linked if G has at least 2k vertices, and for any 2k vertices x 1,x 2, …, x k ,y 1,y 2, …, y k , G contains k pairwise disjoint paths P 1, …, P k such that P i joins x i and y i for i = 1,2, …, k. We say that G is parity-k-linked if G is k-linked and, in addition, the paths P 1, …, P k can be chosen such that the parities of their length are prescribed. Thomassen [22] was the first to prove the existence of a function f(k) such that every f(k)-connected graph is parity-k-linked if the deletion of any 4k-3 vertices leaves a nonbipartite graph.

In this paper, we will show that the above statement is still valid for 50k-connected graphs. This is the first result that connectivity which is a linear function of k guarantees the Erdős-Pósa type result for parity-k-linked graphs.

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Correspondence to Ken-Ichi Kawarabayashi.

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Research partly supported by the Japan Society for the Promotion of Science for Young Scientists, by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research and by Inoue Research Award for Young Scientists.

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Kawarabayashi, KI., Reed, B. Highly parity linked graphs. Combinatorica 29, 215–225 (2009). https://doi.org/10.1007/s00493-009-2178-y

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Mathematics Subject Classification (2000)

  • 05C40
  • 05C83