Tiling Tripartite Graphs with $3$-Colorable Graphs

Abstract

For a fixed integer h ≥ 1, let G be a tripartite graph with N vertices in each vertex class, N divisible by 6h, such that every vertex is adjacent to at least 2N/3 + h − 1 vertices in each of the other classes. We show that if N is sufficiently large, then G can be tiled perfectly by copies of K h,h,h . This extends the work in [19] and also gives a sufficient condition for tiling by any (fixed) 3-colorable graph. Furthermore, we show that this minimum-degree condition is best possible and provide very tight bounds when N is divisible by h but not by 6h.1 and Yuster [1,2] obtained results on H-tiling for arbitrary H. Their results were later improved by various researchers [15,12,21,18].In this paper, we consider multipartite tiling, which restricts G to be an r-partite graph. When r = 2, The König-Hall Theorem (e.g. see [3]) answers the 1-factor problem for bipartite graphs. Wang [24,25] considered K s,s -factors in bipartite graphs for all s > 1, the second author [26] gave the best possible minimum degree condition for this problem.

Minimum degree thresholds for bipartite graph tiling

“…Update G 1 , G 2 as G[A 1 ,B 2 ] and G[B 1 ,Ã 2 ], respectively. Since both v(G 1 ) and v(G 2 ) are divisible by h, we have |A 1 | = m 1 w+s, |B 1 | = m 2 w+t, |Ã 2 | = m 2 u−t, |B 2 | = m 1 u−s (17) for some integers m 1 , m 2 , s, t. Without loss of generality, assume that m 1 ≥ m 2 and consequently t ≥ s. Let c 0 = h+w−1. We separate the cases when t ≤ c 0 and t>c 0 .…”

On multipartite Hajnal–Szemerédi theorems

“…In summary, by (18), (22) and (23), we have (1 − 2kα ′′ ) n k ≤ |A ij | ≤ (1 + 2α ′′ ) n k for all i and j. In order to make k j=1 A ij a partition of V i , we move the vertices of V ′ 1 to A 11 and the vertices of A i0 to A i2 for 2 ≤ i ≤ k. Since |V ′ 1 | < 2αn and |A i0 | ≤ (k + 1)αn, we have ||A ij |− n k | ≤ 2kα ′′ n k after moving these vertices.…”

Tiling Tripartite Graphs with 3-Colorable Graphs: The Extreme Case

“…First, we need a lemma (Lemma 2.1 in [24]) which permits sparse tripartite graphs with no triangles and with no quadrilaterals in its natural bipartite subgraphs:…”

“…The lower bound for f (N, h) in Theorem 3 is due to two constructions, one which is from [24] and another which is similar. They are stated in Proposition 1, and proven in Section 2.…”

“…Theorem 4 ( [24]) Given any positive integer h and any α > 0, there exists an ε > 0 and an integer N 0 such that whenever N ≥ N 0 , and h divides N , the following occurs: If G ∈ G 3 (N ) satisfies δ * (G) ≥ (2/3 − ε)N , then either G contains a K h,h,h -factor or G is α-extremal.…”