Immersing small complete graphs

Abstract

Following in the spirit of the Hadwiger and Hajós conjectures, Abu-Khzam and Langston have conjectured that every k-chromatic graph contains an immersion of K k . They proved this for k ≤ 4. Much before that, Lescure and Meyniel [10] obtained a stronger result that included also the values k = 5 and 6, by proving that every simple graph of minimum degree k − 1 contains an immersion of K k . They noted that they also have a proof of the same result for k = 7 but have not published it due to the length of the proof. We give a simple proof of this result. This, in particular, proves the conjecture of Abu-Khzam and Langston for every k ≤ 7.

A minimum degree condition forcing complete graph immersion

“…Average degree Ω(t √ log t) forces K t as a minor (Kostochka [12], Thomason [20]), and this bound is best possible. For subdivisions, an old conjecture of Mader and Erdős-Hajnal, which was eventually proved by Bollobás and Thomason [3] and independently by Komlós and Szemerédi [11], says that there is a constant c such that every graph with average degree at least ct 2 contains a subdivision of K t , and this is tight apart from the constant c. The corresponding extremal problem for complete graph immersions has been proposed in [6]. Problem 1.2 Let t be a positive integer.…”

“…One of the outcomes of [6] is that Conjecture 1.1 holds for every k ≤ 7. Although the results of this paper are not directly related to Conjecture 1.1, they provide some evidence that supports this conjecture.…”

“…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.…”

Complete graph immersions and minimum degree

“…Conjecture 1 is known to be true for t ≤ 7; for t ≤ 4 the arguments are trivial, for 5 ≤ t ≤ 7 it was proven by DeVos et al [6] and independently, by Lescure and Meyniel [9] (their proof of t = 7 case is not published). In all mentioned cases the authors actually prove stronger statements, showing that only a lower bound on the minimum degree is required to ensure the existence of the immersion.…”

“…The idea of considering this function f (t) was first proposed in [6], as the natural analogue of classical results showing that large average degree (equivalently, large minimum degree) in a graph implies a K t -minor or K t -topological minor containment. To this end, it is known that average degree Ω(t √ log t) in a graph forces a K t -minor and this bound is tight (Kostochka [14] and Thomason [16]).…”