Transversal tilings in k-partite graphs without large holes

We show that for any constant $μ>0$ and $k\ge 3$, there exists $α>0$ such that the following holds for sufficiently large $n \in \mathbb{N}$. If $G=(V_{1},\ldots,V_{k},E)$ is a spanning subgraph of the $n$-blow-up of $K_{k}$ with ${δ^}(G)\geq (\frac{1}{2}+μ) n$ and $α^{k-1}(G)<αn$, then $G$ has a transversal $K{k}$-factor. Moreover, the bound $\frac{1}{2}$ is asymptotically tight for the case (k=3). In addition, we show that if $k\ge 4$, $G=(V_{1},\ldots,V_{k},E)$ is a spanning subgraph of the $n$-blow-up of $C_{k}$ with ${δ^}(G)\ge (\frac{2}{k}+μ) n$, and $α^{2}(G)<αn$, then $G$ has a transversal $C{k}$-factor. This extends a recent result of Han, Hu, Ping, Wang, Wang and Yang.

💡 Research Summary

The paper investigates transversal tilings in multipartite graphs under the additional restriction that the graph contains no large partite holes. The authors focus on two classic tiling problems: transversal K_k‑factors in the n‑blow‑up of the complete graph K_k, and transversal C_k‑factors in the n‑blow‑up of the cycle C_k.

The main contributions are two theorems that substantially lower the minimum partite‑degree thresholds required for the existence of such tilings, provided that the size of the largest (k‑1)-partite hole (for K_k‑factors) or the largest 2‑partite hole (for C_k‑factors) is sublinear in n.

Theorem 1.3 states that for any constant μ>0 and integer k≥3 there exists a constant α>0 such that, for sufficiently large n, any spanning subgraph G of the n‑blow‑up of K_k satisfying

• minimum bipartite degree δ⁎(G) ≥ (½ + μ)n, and
• α⁎_{k‑1}(G) < αn,
  contains a transversal K_k‑factor. The bound ½ is shown to be asymptotically tight when k=3, via a construction that yields a graph with δ⁎(G)≈½n but no transversal triangle factor. For k≥4 the exact threshold remains open, but the result already improves on the classical multipartite Hajnal–Szemerédi bound (1 − 1/k)n.

Theorem 1.6 extends the analysis to cycles. For any integer k≥4 and any constant δ>2/k there exists α>0 such that, for large n, any spanning subgraph G of the n‑blow‑up of C_k with

• δ⁎(G) ≥ (2/k + δ)n, and
• α⁎_2(G) < αn,
  admits a transversal C_k‑factor. This strengthens a recent result of Han et al., which required k to be a multiple of 4 and a larger degree condition. The authors note that without the α⁎_2(G) condition the best known bound is δ⁎(G) ≥ (1 + 1/k)n/2, and conjecture that the optimal asymptotic bound should be n/k (the space barrier).

The proofs rely on the absorbing method, a powerful technique for constructing spanning structures. The authors develop a lattice‑based absorbing framework, originally introduced by Montgomery and later refined by Nenadov–Pehoja, to guarantee the existence of a small balanced absorbing set R. This set can absorb any leftover balanced vertex set of size at most ξn.

The construction proceeds as follows:

1. Absorbing set – Using Lemma 2.3 (for K_k) and Lemma 2.4 (for C_k), they show that every transversal k‑set possesses linearly many (F,t)-absorbers, where F is K_k or C_k. By a probabilistic selection argument, a global absorbing set R of size O(γn) is obtained.
2. Almost‑perfect tiling – Applying the Szemerédi Regularity Lemma, the graph is reduced to a bounded‑size multipartite multigraph. Lemma 2.5 (K_k) and Lemma 2.6 (C_k) guarantee a transversal tiling covering all but at most ζn vertices of the reduced graph, which lifts to an almost‑perfect tiling of the original graph.
3. Absorption – The uncovered vertices U (|U|≤ζn) are merged with the absorbing set R. By definition of R, G