A minimum degree condition forcing complete graph immersion

• Main Authors: Devos, Matt (Author), Fox, Jacob (Contributor), McDonald, Jessica (Author), Mohar, Bojan (Author), Scheide, Diego (Author), Dvorak, Zdenek (Author)
• Other Authors: Massachusetts Institute of Technology. Department of Mathematics (Contributor)
• Format: Article
• Language: English
• Published: Springer-Verlag, 2015-01-14T16:40:07Z.
• Subjects: Article
• Online Access: Get fulltext

An immersion of a graph H into a graph G is a one-to-one mapping f: V (H) → V (G) and a collection of edge-disjoint paths in G, one for each edge of H, such that the path P [subscript uv] corresponding to edge uv has endpoints f(u) and f(v). The immersion is strong if the paths P [subscript uv] are internally disjoint from f(V (H)). It is proved that for every positive integer Ht, every simple graph of minimum degree at least 200t contains a strong immersion of the complete graph K [subscript t]. For dense graphs one can say even more. If the graph has order n and has 2cn [superscript 2] edges, then there is a strong immersion of the complete graph on at least c [superscript 2] n vertices in G in which each path P [subscript uv] is of length 2. As an application of these results, we resolve a problem raised by Paul Seymour by proving that the line graph of every simple graph with average degree d has a clique minor of order at least cd [superscript 3/2], where c>0 is an absolute constant. For small values of t, 1≤t≤7, every simple graph of minimum degree at least t−1 contains an immersion of K [subscript t] (Lescure and Meyniel [13], DeVos et al. [6]). We provide a general class of examples showing that this does not hold when t is large.
Simons Foundation (Fellowship)
National Science Foundation (U.S.) (Grant DMS-1069197)
NEC Corporation (MIT Award)