In 1989, Lescure and Meyniel proved, for d=5,6, that every d-chromatic
graph contains an immersion of Kdβ, 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 d=7. 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 dβ1. DeVos,
Dvo\v{r}\'ak, Fox, McDonald, Mohar, and Scheide show the stronger conjecture
that a graph with minimum degree dβ1 has an immersion of Kdβ fails for
d=10 and dβ₯12 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 Kdβ.
In this paper we show that the stronger conjecture is false for d=8,9,11
and give infinite families of examples with minimum degree dβ1 and chromatic
number dβ3 or dβ2 that do not contain an immersion of Kdβ. Our examples
can be up to (dβ2)-edge-connected. We show, using Haj\'os' Construction, that
there is an infinite class of non-(dβ1)-colorable graphs that contain an
immersion of Kdβ. We conclude with some open questions, and the conjecture
that a graph G with minimum degree dβ1 and more than
1+m(d+1)β£V(G)β£β vertices of degree at least md has an immersion of
Kdβ