Spanning tight components in 4-uniform hypergraphs

Abstract

We prove that every $n$-vertex 4-uniform hypergraph with minimum codegree at least $\lfloor n/4 \rfloor$ has a spanning tight component. This is tight, and it settles the 4-uniform case of a conjecture of Illingworth, Lang, Müyesser, Parczyk, and Sgueglia.

Figures (4)

• Figure 1: A schematic depiction of the unique $5$-configuration and the unique $6$-configuration with no pair of vertices lying in edges of $4$ distinct colours. The $5$-configuration has as its colour classes the five rotations of the tight path in (a). The $6$-configuration has as one of its colour classes the tight $5$-cycle on the right in (b), while the remaining five classes correspond to the five rotations of the tight path on the left.
• Figure 2: The structures involved in the proof of \ref{['lem:symmetric_four']}.
• Figure 3: The structure involved in the proof of \ref{['lem:UB_colours']}.
• Figure 4: An example of a $(8m+1)$-abundant colouring of $K_{32m+7}$ using $6$ colours. The vertex set is partitioned into seven clusters, one of size $8m+1$ and the others of size $4m+1$. Inside each cluster of size $4m+1$, the two colours depicted in the figure are used, each inducing a $2m$-regular graph spanning the cluster; the same applies for the larger cluster, except that in this case the colour classes are $4m$-regular. The edges between the clusters behave according to the diagram.

Theorems & Definitions (27)

• Conjecture 1.1: spanningspheres
• Theorem 1.2
• Theorem 1.3
• proof : Proof of the equivalence of \ref{['thm:main']} and \ref{['thm:main_colouring']}.
• Lemma 3.1
• Definition 3.2: $r$-configuration
• Lemma 3.3
• proof
• proof : Proof of \ref{['thm:main_colouring']}
• proof