Abstract:

Let A and B be two sets, the symmetry difference of A and B is a set consisting of all elements not belonging to $A\cap B$ in $A\cup B$, denoted by $A\Delta B$. A hypergraph is called a cancellative hypergraph, if it contains no three distinct edges A, B, and C, such that $A\Delta B\subset C$. In fact, a cancellative3-uniform hypergraph contains neither $F_4=\{abc, abd, bcd\}$ nor $F_5=\{abc, abd, cde\}$ as its subgraphs. Bollobás (1974) determined the maximum number of edges in a cancellative3-uniform hypergraph, and got that only the balanced complete3-partite3-uniform hypergraph achieved the maximum number of edges in a cancellative3-uniform hypergraph. Furthermore, Keevash and Mubayi (2004) determined that only the balanced complete3-partite3-uniform hypergraph achieved the maximum number of edges in a3-uniform hypergraph containing no copy of $F_5$. Let $\mathcal{H}$ be a hypergraph, and W be a nonempty subset of $V(\mathcal{H})$. If every edge in $\mathcal{H}$ contains exactly one vertex in W, then we call W an independent transversal of $\mathcal{H}$. In this paper, we determine the maximump-spectral radius of a cancellative3-uniform hypergraph with an independent transversal. Furthermore, if $p>2$, we get that only the balanced complete3-partite3-uniform hypergraph achieved the maximump-spectral radius of a cancellative3-uniform hypergraph with an independent transversal.

Key words: cancellative 3-uniform hypergraph, p-spectral radius, independent transversal