Abstract
As a variation on the $t$-Equal Union Property ($t$-EUP) introduced by Lindström, we introduce the $t$-Equal Valence Property ($t$-EVP) for hypergraphs: a hypergraph satisfies the $t$-EVP if there are $t$ pairwise edge-disjoint subhypergraphs such that for each vertex $v$, the degree of $v$ in all $t$ subhypergraphs is the same. In the $t$-EUP, the subhypergraphs just have the same sets of vertices with positive degree. For both the $2$-EUP and the $2$-EVP, we characterize the graphs satisfying the property and determine the maximum number of edges in a graph not satisfying it. We also study the maximum number of edges in both $k$-uniform and general hypergraphs not satisfying the $t$-EVP.
Highlights
We consider conditions for a hypergraph to have t edge-disjoint subhypergraphs whose local behavior at vertices is somehow “the same”
We introduce the t-Equal Valence Property (t-EVP), more restrictive than the t-Equal Union Property (t-EUP)
We study the extremal problem posed by Lindstrom and the analogous extremal problem for equal valence, considering the restrictions to k-uniform hypergraphs
Summary
We consider conditions for a hypergraph to have t edge-disjoint subhypergraphs whose local behavior at vertices is somehow “the same”. A hypergraph H satisfies the t-Equal Union Property (t-EUP) if H has t edge-disjoint distinct subhypergraphs H1, . The Fano plane is a 3-uniform hypergraph with seven vertices and seven edges that does not satisfy the 2-EUP, thereby providing another sharpness example. A hypergraph H satisfies the t-Equal Valence Property (t-EVP) if H has t edge-disjoint distinct subhypergraphs H1, . A projective plane of order q is a (q + 1)-uniform hypergraph with q2 + q + 1 points and q2 + q + 1 edges (called lines), in which every two lines have exactly one common the electronic journal of combinatorics 21(1) (2014), #P1.62 point, every two points appear in exactly one common line, and there exist four points among which no three appear in one line. Let U(n, t) denote the maximum number of edges in an n-vertex hypergraph not satisfying the t-EUP. Let Uk(n, t) denote the maximum number of edges in a k-uniform n-vertex hypergraph not satisfying the t-EUP. Since the usage of edges incident to v is the same in each Hi, ignoring those edges contributes the same degree from all t subgraphs at each of the remaining vertices
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