A minimum degree condition forcing complete graph immersion

Abstract

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 uv corresponding to edge uv has endpoints f(u) and f(v). The immersion is strong if the paths P 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 t . For dense graphs one can say even more. If the graph has order n and has 2cn 2 edges, then there is a strong immersion of the complete graph on at least c 2 n vertices in G in which each path P 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 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 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.

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Correspondence to Matt Devos.

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Supported in part by an NSERC Discovery Grant (Canada) and a Sloan Fellowship.

Supported by the Center of Excellence — Inst. for Theor. Comp. Sci., Prague, project P202/12/G061 of Czech Science Foundation.

Supported by a Simons Fellowship, NSF grant DMS-1069197, and by an MIT NEC Corporation Award.

Supported by an NSERC Postdoctoral fellowship.

Supported in part by an NSERC Discovery Grant (Canada), by the Canada Research Chair program, and by the Research Grant P1-0297 of ARRS (Slovenia).

Postdoctoral fellowship at Simon Fraser University, Burnaby.

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Devos, M., Dvořák, Z., Fox, J. et al. A minimum degree condition forcing complete graph immersion. Combinatorica 34, 279–298 (2014). https://doi.org/10.1007/s00493-014-2806-z

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Mathematics Subject Classification (2000)

  • 05C35
  • 05C83