Embedding clique-factors in graphs with low $\ell$-independence number

Abstract

The following question was proposed by Nenadov and Pehova and reiterated by Knierim and Su: Given integers , r and n with n ∈ rN, is it true that every n-vertex graph G with δ(G) ≥ max{ 1 2 , r− r }n + o(n) and α (G) = o(n) contains a K r -factor? We give a negative answer for the case when ≥ 3r 4 by giving a family of constructions using the so-called cover thresholds and show that the minimum degree condition given by our construction is asymptotically best possible. That is, for all integers r, with r > ≥ 3 4 r and µ > 0, there exist α > 0 and N such that for every n ∈ rN with n > N , every n-vertex graph G with δ(G) ≥ 1 2− (r−1) + µ n and α (G) ≤ αn contains a K r -factor. Here (r − 1) is the Ramsey-Turán density for K r−1 under the -independence number condition.

Clique-factors in graphs with sublinear $\ell$-independence number

“…Proposition 2.4 (see Proposition 2.6 in [8]). Given a constant ε > 0 and integers m, t with t < 1 2ε , let G be an n-vertex graph and V 1 , V 2 , .…”

“…[25,34] Is it true that for every r, ∈ N with r ≥ and µ > 0, there exists α > 0 such that for sufficiently large n ∈ rN, every n-vertex graph G with δ(G) ≥ max{ 1 2 +µ, r− r +µ}n and α (G) ≤ αn contains a K r -factor? Very recently, a result of Chang et al [8] determines the asymptotically optimal minimum degree condition for ≥ 3 4 r, which solves Problem 1.2 for this range, and also provides a negative answer to Problem 1.3.…”

“…To have a K r -factor in G, a naive necessary condition is that every vertex v ∈ V (G) is covered by a copy of K r in G. The cover threshold has been first discussed in [20] and appeared in a few different contexts [8,9,12]. 1 2− * (r−1,f ) for the function f (n) as in Theorem 1.6 and a similar construction can be found in [8]. Given integers r, and constants ε, α, x ∈ (0, 1), we construct (for large n ∈ rN) an n-vertex graph G by…”

“…We include a proof of Lemma 4.6 as the reference [8] is still under review. We start with the following result for edge-weighted graphs.…”

“…Lemma B.1. [8] Let µ, c be any positive constants. Then there exist integers t 0 = t 0 (µ) and k 0 = k 0 (µ) such that the following holds for k ≥ k 0 and t ≥ t 0 .…”