Transversals in 4-Uniform Hypergraphs

Abstract

The transversal number τ (H) of a hypergraph H is the minimum number of vertices that intersect every edge of H. A linear hypergraph is one in which every two distinct edges intersect in at most one vertex. A k-uniform hypergraph has all edges of size k. Very few papers give bounds on the transversal number for linear hypergraphs, even though these appear in many applications, as it seems difficult to utilise the linearity in the known techniques. This paper is one of the first that give strong non-trivial bounds on the transversal number for linear hypergraphs, which is better than for non-linear hypergraphs. It is known that τ (H) ≤ (n + m)/(k + 1) holds for all k-uniform, linear hypergraphs H when k ∈ {2, 3} or when k ≥ 4 and the maximum degree of H is at most two. It has been conjectured (at several conference talks) that τ (H) ≤ (n + m)/(k + 1) holds for all k-uniform, linear hypergraphs H. We disprove the conjecture for large k, and show that the best possible constant c k in the bound τ (H) ≤ c k (n + m) has order ln(k)/k for both linear (which we show in this paper) and non-linear hypergraphs. We show that for those k where the conjecture holds, it is tight for a large number of densities if there exists an affine plane AG(2, k) of order k ≥ 2. We raise the problem to find the smallest value, k min , of k for which the conjecture fails. We prove a general result, which when applied to a projective plane of order 331 shows that k min ≤ 166. Even though the conjecture fails for large k, our main result is that it still holds for k = 4, implying that k min ≥ 5. The case k = 4 is much more difficult than the cases k ∈ {2, 3}, as the conjecture does not hold for general (non-linear) hypergraphs when k = 4. Key to our proof is the completely new technique of the deficiency of a hypergraph introduced in this paper.Claim H.6: The case |X| ≥ 2 and |∂(X)| ≥ 4 cannot occur.Proof of Claim H.6: Note that 8|X 4 |+5|X 14 |+4|X 11 |+|X 21 | ≤ 8|X| ≤ 13|X|−6|X ′ |−10, as |X| ≥ 2 and |X ′ | = 0. If H ′ is linear, then we are obtain a contradiction to Claim H.4(a), and if H ′ is not linear we obtain a contradiction to Claim H.5. (✷) Claim H.7: The case |X| ≥ 3 and |∂(X)| = 3 cannot occur.Proof of Claim H.7: Note that 8|X 4 | + 5|X 14 | + 4|X 11 | + |X 21 | ≤ 8|X| ≤ 13|X| − 6|X ′ | − 9, as |X| ≥ 3 and |X ′ | = 1. If H ′ is linear, then we are obtain a contradiction to Claim H.4(b), and if H ′ is not linear we obtain a contradiction to Claim H.5. (✷) Claim H.8: The case |X| = 2 and |∂(X)| = 3 cannot occur.Proof of Claim H.8: Suppose, to the contrary, that |X| = 2 and |∂(X)| = 3. Then, |E * (X)| = 3 and |X ′ | = 1. If H ′ is not linear, then we obtain a contradiction by Claim H.5(a), as 8|X 4 | + 5|X 14 | + 4|X 11 | + |X 21 | ≤ 16 and 13|X| − 6|X ′ | − 3 = 17. Therefore, H ′ is linear.

Dominating vertex covers: the vertex-edge domination problem

“…(B) Given any B ′ ∈ B containing a 2-edge {u, v}, define B as follows. Using Theorem 2, we can prove the following result, which is implicit in [3]; therefore we include a short proof for completeness. Proof.…”

“…In order to state the main result from [3], we need to define a particular class of hypergraphs B. Let B be the class of bad hypergraphs defined as exactly those that can be generated using the operations (A)-(D) below.…”

Transversals in regular uniform hypergraphs

“…The authors in [11] proved that if For ≥ p 1 an integer and for k p = 3 and ≥ i 1, we define Q k i , as the hypergraph obtained from Q i 3, by duplicating each vertex p times, that is, we replace each vertex by p identical copies of the vertex. The hypergraph Q 6,13 is illustrated in Figure 4.…”

“…A transversal of $$H$$ of size $$\tau (H)$$ is called a $$\tau $$‐ transversal of $$H$$. A small sample of papers on transversals in hypergraphs can be found, for example, in [1, 3–7, 9–11, 13–17, 20].…”

Every 4-regular 4-uniform hypergraph has a 2-coloring with a free vertex

“…The degree of a vertex v in H, denoted d H (v) or simply by d(v) if H is clear from the context, is the number of edges of H which contain v. The hypergraph H is k-regular if every vertex has degree k in H. For k ≥ 2, let H k denote the class of all k-uniform k-regular hypergraphs. The class H k has been widely studied, both in the context of solving problems on total domination as well as in its own right, see for example [1,3,4,5,10].…”