Linkage for the diamond and the path with four vertices

  • Authors:

  • M. N. Ellingham;Michael D. Plummer;Gexin Yu

  • Affiliations:

  • Department of Mathematics, 1326 Stevenson Center Vanderbilt University, Nashville Tennessee, 37240;Department of Mathematics, 1326 Stevenson Center Vanderbilt University, Nashville Tennessee, 37240;Department of Mathematics, College of William and Mary, Williamsburg, 8795Virginia23187

  • Venue:
  • Journal of Graph Theory
  • Year:
  • 2012

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Abstract

Given graphs G and H, we say G is H-linked if for every injective mapping ℓ: V(H)→V(G), we can find a subgraph H′ of G that is a subdivision of H with ℓ(v) being the vertex of H′ corresponding to each vertex v of H. In this article, we prove two results on H-linkage for 4-vertex graphs H. Goddard showed that 4-connected planar triangulations are 4-ordered, or in other words C4-linked. We strengthen this by showing that 4-connected planar triangulations are (K4−e)-linked. X. Yu characterized certain graphs related to P4-linkage. We use his characterization to show that every 7-connected graph is P4-linked, and to construct 6-connected graphs that are not P4-linked. © 2011 Wiley Periodicals, Inc. J Graph Theory, © 2012 Wiley Periodicals, Inc. (Contract grant sponsor: U.S. National Security Agency; Contract grant numbers: H98230-06-1-0051; H98230-09-1-0065 (to M. N. E.); Contract grant sponsor: NSF; Contract grant numbers: DMS-0652306; DMS-0852452 (to G. Y.). The United States Government is authorized to reproduce and distribute reprints notwith-standing any copyright notation herein.)