On Domination and 2-packing Numbers in Intersecting Linear Systems

A linear system is a pair \((P,\mathcal{L})\) where \(\mathcal{L}\) is a finite family of subsets on a finite ground set \(P\) such that any two subsets of \(\mathcal{L}\) share at most one element. Furthermore, if for every two subsets of \(\mathcal{L}\) share exactly one element, the linear system is called intersecting. A linear system \((P,\mathcal{L})\) has rank \(r\) if the maximum size of any element of \(\mathcal{L}\) is \(r\). By \(\gamma(P,\mathcal{L})\) and \(\nu_2(P,\mathcal{L})\) we denote the size of the minimum dominating set and the maximum 2-packing of a linear system \((P,\mathcal{L})\), respectively. It is known that any intersecting linear system \((P,\mathcal{L})\) of rank \(r\) is such that \(\gamma(P,\mathcal{L})\leq r-1\). Li et al. in [S. Li, L. Kang, E. Shan and Y. Dong, The finite projective plane and the 5-Uniform linear intersecting hypergraphs with domination number four, Graphs and 34 Combinatorics (2018) , no.~5, 931–945.] proved that every intersecting linear system of rank 5 satisfying \(\gamma(P,\mathcal{L})=4\) can be constructed from a 4-uniform intersecting linear subsystem \((P^\prime,\mathcal{L}^\prime)\) of the projective plane of order 3 satisfying \(\tau(P^\prime,\mathcal{L}^\prime)=\nu_2(P^\prime,\mathcal{L}^\prime)=4\), where \(\tau(P^\prime,\mathcal{L}^\prime)\) is the transversal number of \((P^\prime,\mathcal{L}^\prime)\). In this paper, we give an alternative proof of this result given by Li et al., giving a complete characterization of these 4-uniform intersecting linear subsystems. Moreover, we prove a general case, that is, we prove if $q$ is an odd prime power and \((P,\mathcal{L})\) is an intersecting linear system of rank \((q+2)\) satisfying \(\gamma(P,\mathcal{L})=q+1\), then this linear system can be constructed from a spanning \((q+1)\)-uniform intersecting linear subsystem \((P^\prime,\mathcal{L}^\prime)\) of the projective plane of order \(q\) satisfying \(\tau(P^\prime,\mathcal{L}^\prime)=\nu_2(P^\prime,\mathcal{L}^\prime)=q+1\).

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