# A minimum degree condition forcing complete graph immersion

- 243 Downloads
- 15 Citations

## 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 *H*t, every simple graph of minimum degree at least 200*t* 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 2*cn* ^{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.

## Mathematics Subject Classification (2000)

05C35 05C83## Preview

Unable to display preview. Download preview PDF.

## References

- [1]
- [2]N. Alon, M. Krivelevich, B. Sudakov: Turán numbers of bipartite graphs and related Ramsey-type questions,
*Combin. Probab. Comput.***12**(2003), 477–494.CrossRefzbMATHMathSciNetGoogle Scholar - [3]B. Bollobás, A. Thomason: Proof of a conjecture of Mader, Erdős and Hajnal on topological complete subgraphs,
*European J. Combin.***19**(1998), 883–887.CrossRefzbMATHMathSciNetGoogle Scholar - [4]H. D. Booth, R. Govindan, M. A. Langston and S. Ramachandramurthis: Sequential and parallel algorithms for K4-immersion testing,
*J. Algorithms***30**(1999), 344–378.CrossRefzbMATHMathSciNetGoogle Scholar - [5]P. A. Catlin: A bound on the chromatic number of a graph,
*Discrete Math.***22**(1978), 81–83.CrossRefMathSciNetGoogle Scholar - [6]M. DeVos, K. Kawarabayashi, B. Mohar and H. Okamura: Immersing small complete graphs,
*Ars Math. Contemp.***3**(2010), 139–146.zbMATHMathSciNetGoogle Scholar - [7]P. Erdős: Problems and results in graph theory and combinatorial analysis, in:
*Graph Theory and Related Topics*(Proc. Conf. Waterloo, 1977), Academic Press, New York, 1979, 153–163.Google Scholar - [8]M. R. Fellows and M. A. Langston: Nonconstructive tools for proving polynomial-time decidability,
*J. ACM***35**(1998), 727–738.CrossRefMathSciNetGoogle Scholar - [9]J. Fox and B. Sudakov: Dependent random choice,
*Random Structures and Algorithms***38**(2011), 68–99.CrossRefzbMATHMathSciNetGoogle Scholar - [10]H. Hadwiger: Über eine Klassi kation der Streckenkomplexe,
*Vierteljahrsschr. Naturforsch. Ges. Zürich***88**(1943), 133–142.MathSciNetGoogle Scholar - [11]J. Komlós, E. Szemerédi: Topological cliques in graphs. II,
*Combin.***Probab. Comput. 5**(1996), 79–90.Google Scholar - [12]A. Kostochka: Lower bound of the Hadwiger number of graphs by their average degree,
*Combinatorica***4**(1984), 307–316.CrossRefzbMATHMathSciNetGoogle Scholar - [13]F. Lescure, H. Meyniel: On a problem upon configurations contained in graphs with given chromatic number, Graph theory in memory of G. A. Dirac (Sandbjerg, 1985), 325–331, Ann. Discrete Math. 41, North-Holland, Amsterdam, 1989.Google Scholar
- [14]L. Lovász, M. D. Plummer:
*Matching Theory*, AMS Chelsea Publishing, Providence, RI, 2009.zbMATHGoogle Scholar - [15]C. St. J. A. Nash-Williams: Edge-disjoint spanning trees of finite graphs,
*J. London Math. Soc.***36**(1961), 445–450.CrossRefzbMATHMathSciNetGoogle Scholar - [16]N. Robertson and P. D. Seymour: Graph minors. XX. Wagner’s conjecture,
*J. Combin. Theory Ser. B***92**(2004), 325–357.CrossRefzbMATHMathSciNetGoogle Scholar - [17]N. Robertson and P. D. Seymour: Graph Minors XXIII, Nash-Williams’ immersion conjecture,
*J. Combin. Theory Ser. B***100**(2010), 181–205.CrossRefzbMATHMathSciNetGoogle Scholar - [18]N. Robertson, P. D. Seymour and R. Thomas: Hadwiger’s conjecture for
*K*_{6}-free graphs,*Combinatorica***13**(1993), 279–361.CrossRefzbMATHMathSciNetGoogle Scholar - [19]P. Seymour: personal communication.Google Scholar
- [20]A. Thomason: The extremal function for complete minors,
*J. Combin. Theory Ser. B***81**(2001), 318–338.CrossRefzbMATHMathSciNetGoogle Scholar - [21]W. T. Tutte: On the problem of decomposing a graph into
*n*connected factors,*J. London Math. Soc.***36**(1961), 221–230.CrossRefzbMATHMathSciNetGoogle Scholar