Large Immersions in Graphs with Independence Number 3 and 4

The analogue of Hadwiger's conjecture for the immersion order, a conjecture independently posed by Lescure and Meyniel, and by Abu-Khzam and Langston, states that every graph G which does not contain the complete graph K_t+1 as an immersion satisfies χ(G) ≤ t. If true, this conjecture would imply that every graph with n vertices and independence number α contains K_{⌈n/3.5α-c⌉} as an immersion (and if α = 2, the two statements are known to be equivalent). The immersion conjecture has been tackled with more success than its graph minors counterpart: not only is a linear upper bound known for the chromatic number of K_t+1-immersion-free graphs, but the best bound currently known is very close to optimal. Namely, the currently best bound in this respect is due to Gauthier, Le and Wollan, who recently proved that every graph not containing K_t+1 as an immersion satisfies χ(G) ≤ 3.54t + 7. Their result implies that any graph with n vertices and independence number α contains K_2⌈n/5⌉ as an immersion, where c < 1.98. Moreover, the same authors prove that every graph of independence number 2 contains K_⌊n/α⌋ as an immersion. We show that any graph with n vertices and independence number 3 contains a clique immersion on at least ⌈2n/9⌉ vertices, and any graph with n vertices and independence number 4 contains a clique immersion on at least ⌈4n/27⌉ -1 vertices. Thus, comparing to the bound from above, in both cases we roughly double the size of the immersion obtained.