Constructing Graphs with No Immersion of Large Complete Graphs

Abstract

AbstractIn 1989, Lescure and Meyniel proved, for , that every d‐chromatic graph contains an immersion of , and in 2003 Abu‐Khzam and Langston conjectured that this holds for all d. In 2010, DeVos, Kawarabayashi, Mohar, and Okamura proved this conjecture for . In each proof, the d‐chromatic assumption was not fully utilized, as the proofs only use the fact that a d‐critical graph has minimum degree at least . DeVos, Dvořák, Fox, McDonald, Mohar, and Scheide show the stronger conjecture that a graph with minimum degree has an immersion of fails for and with a finite number of examples for each value of d, and small chromatic number relative to d, but it is shown that a minimum degree of 200d does guarantee an immersion of .In this paper, we show that the stronger conjecture is false for and give infinite families of examples with minimum degree and chromatic number , , or that do not contain an immersion of . Our examples can be up to ‐edge‐connected. We show that two of Hajós' operations preserve immersions of . We conclude with some open questions, and the conjecture that a 2‐edge‐connected graph G with minimum degree and more than vertices of degree at least has an immersion of .